DEFINABLE AND CONTRACTIBLE CONTRACTS
MICHAEL PETERS AND BALÁZS SZENTES
Abstrat.
This paper analyzes Bayesian normal form games in whih players write ontrats
that ondition their ations on the ontrats of the other players. These ontrats are required
to be representable in a formal language. This is aomplished by onstruting ontrats whih
are denable funtions of the Godel ode of every other player's ontrat. We provide a omplete
haraterization of the set of alloations supportable as pure strategy Bayesian equilibrium of this
ontrating game. When information is omplete, this haraterization provides a folk theorem.
In general, the set of supportable alloations is smaller than the set supportable by a entralized
mehanism designer.
1.
Self Referential Strategies and Reiproity in Stati Games
In this paper we haraterize the alloation rules attainable by players in a Bayesian game when
they have the ability to ommit themselves by writing ontrats that ondition their ommitments
on other players' ontrats.
The idea that ontrats might ondition on other ontrats is not new in eonomis. The best
known expression of this idea is well known in the industrial organization literature (e.g. [11℄) as
the 'meet the ompetition' lause in whih one rm ommits itself to lower its prie when any of its
ompetitors does. A similar idea appears in trade theory as the priniple of
reiproity
([2℄). This
takes the form of trade agreements like GATT that require ountries to math tari uts by other
ountries. Finally, tax treaties sometimes have this avor - for example, out of state residents who
work in Pennsylvania are exempt from Pennsylvania tax as long as they live in a state that has a
'reiproal' agreement that exempts out of state residents (presumably from Pennsylvania) from
1
state taxes.
All of these approahes are used to support ooperative outomes in stati games. We extend this
approah to games with inomplete information. We provide a full haraterization of alloations
supportable as ontrat equilibrium. In partiular, we show the limits of the 'ontrats on ontrats'
approah by providing alloations supportable by a mehanism designer whih annot be supported
as equilibrium with ontratible ontrats. We also use our Theorem to provide something that
looks like a folk theorem for a restrited environment.
The diulty with extending the older literature is that the oneptual and tehnial tools
developed there an only be used in any but the simplest problems. The meet the ompetition
Version - January 22, 2009.
1http://www.revenue.state.pa.us/revenue/wp/view.asp?A=238&Q=244681
1
2
MICHAEL PETERS AND BALÁZS SZENTES
argument, for example, is extremely stylized. The Stakleberg leader, all it rm A, oers to sell at
a very high prie provided its ompetitor, rm B, also oers that high prie in the seond round. If
B
in the seond round oers any prie below the highest prie,
ost. If
B
A ommits itself
to sell at marginal
believes this ommitment, then one best reply is to set the highest prie.
If the rms move simultaneously, then the logi of the argument beomes louded.
ertainly write a ontrat that ommits it to a high prie if
suppose that
to
A's
B
A
ould
sets the same high prie. However
B 's strategy is simply to set this high prie and that for some reason this is a best reply
ontrat. Then
outome, rm
B
A
should deviate and simply underut rm
would have to oer a ontrat similar to
naive argument would suggest that
B
A's
B.
To support the high prie
in order to prevent
should simply oer the same ontrat as
A,
A's
deviation. A
a high prie if
A
sets a high prie, and marginal ost otherwise. Casually, two outomes seem onsistent with these
ontrats - both rms prie at marginal ost or both rms set the high prie. This seems to violate
a fairly fundamental property of game theory whih is that for eah pair of ations (ontrats
in this ase), there is a unique payo to every player.
atually say what
A
would do if
B
2
More to the point,
A's
ontrat doesn't
oers a ontrat that promises to set a high prie unless
A
sets
a lower prie, et. The speiation of the problem itself seems to be ambiguous about payos.
Generally ontrats that reat to
ations
of other players simply don't make sense. They may
not lead to unambiguous outomes as in the example above. More generally, it is possible that
suh ontrats are simply ontraditory. For example, two rms might write ontrat that ommit
both of them to set a prie that is stritly lower than the other rms prie (or two eonomists
demand ontrats that guarantee that they will both earn more money than anyone else in the
department). To resolve ambiguities and ontraditions in suh ontrat, an outside mediator is
needed to hoose an outome.
This defeats the purpose of using ontrats to deentralize the
underlying alloation problem.
The reiproal tax agreement problem is better behaved, and provides the basis for the argument
we extend below. State
residents of state
A
A wants to exempt residents of state B from state taxes provided B exempts
from taxes. To write the law
'reiproal' agreement with state
A.
A
exempts residents from any state that has a
The question is what exatly is a 'reiproal' agreement. It
is lear enough what the intention is - reate a situation in whih both states take the mutually
beneial ation of exempting one another in a way that eliminates any inentive for either of them
to deviate. As mentioned above, it isn't enough to assume that state
residents of state
State
2
B
A
from tax beause
A
B
unonditionally exempts
would not longer have any inentive to exempt state
has to have a law like the law in state
A,
B.
in other words, a reiproal agreement.
One paper that allows multiple payos to be assoiated with eah array of ations is [12℄ who use this approah
to support equilibrium when it might not otherwise exist.
DEFINABLE AND CONTRACTIBLE CONTRACTS
3
It seems that to resolve this kind of problem one needs to dene the term 'reiproal ontrat'
as follows:
reiproal ontrat
≡

exempt
don't
if the other state oers a reiproal ontrat,
otherwise
This kind of denition is familiar from the Bellman equation in dynami programming where the
value funtion is dened in a self referential way. It is tempting to model this in the following naive
way: start by dening a olletion of ontrats that seem eonomially sensible. For example, it is
reasonable that a state ould write a ontrat that simply xes any tax rate independent of what
the other states do. Let
C
be the set of ontrats that simply x some unonditional tax rate.
Append to this set of feasible ontrats the reiproal ontrat, all it
the set of feasible ontrats as
C ∪ {r}.
r,
dened above. Now model
The reiproal ontrat above is just
r,
while 'otherwise'
means any ontrat with a xed tax rate. Dene a normal form game in whih the strategies are
C ∪ {r}
and delare the outome if both states oer
r
to be
(exempt, exempt).
Then there is an
equilibrium in whih the states mutually exempt (assuming they jointly want to).
We would argue that this is unsatisfatory for a number of reasons. First, it is undesirable to
restrit the set of feasible ontrats in order to support the outome you are looking for.
approah desribed above amounts to little more than saying that
then laiming it is an equilibrium for both states to oer
r.
r
The
is the only feasible ontrat,
A more satisfatory approah is
to dene a set of ations that seem eonomially meaningful, then to allow the broadest set of
ontrats possible. In the same manner that the value funtion emerges endogenously from the
eonomi environment, the reiproal ontrat should be derived from eonomi fundamentals.
Seond, the approah desribed above misses the essene of reiproity whih is the innite
regress involved in self referential objets. A ontrat that makes formal sense is the following:
C
=

exempt
don't
if other State exempts any State who exempts any State who exempts. . .
otherwise
where the statement in the top line is repeated ad innitum.
Arguably, the ontrat
C
is a
reiproal ontrat sine it would exempt any State oering a reiproal ontrat. Yet it simply
isn't feasible under the naive desription given above.
Finally, the ad ho approah desribed above simply isn't exible enough to handle omplex
environments and inomplete information. For example, making the game asymmetri requires ad
ho extension of the approah above. If State
A
is supposed to exempt, while state
B
is supposed
to take some other ation, say 'partly exempt', then to support the right outome, the ontrats
should look something like the following:
reiproal ontratA
≡

exempt
don't
if other State oers reiproal ontratB
exempt otherwise
4
MICHAEL PETERS AND BALÁZS SZENTES
and
reiproal ontratB
≡

partially
don't
exempt if other State oers reiproal ontratA
exempt otherwise
Now the ontrats are not diretly self referential, as is the Bellman equation, instead they are ross
referential. A single self referential or reiproal ontrat simply doesn't go far enough. Furthermore, the ontrats above dene only a single ooperative ation, and use a blanket punishment
for deviations. Desirable or interesting equilibrium alloations may not look like this. For example,
in a general Bayesian game, the most desirable ooperative ation for both players might depend
on information that only one of them has. So the ation that State A wants to take might depend
on the ontrat that B oers. Alternatively, the most eetive punishment for A to impose on B
might depend on ations that other states are taking.
As the number of possibilities inreases,
so does the number of speial words we need to add to our ontrating language to support the
outomes we want.
Our approah avoids these problems. We x a language and require ontrats to be written in
this language. We then show this language already ontains all the speial terms like 'reiproal
ontrat' that we need, even in very rih eonomi environments where simple notions like 'ooperation' do not adequately desribe the alloations we are interested in. The ontrating language
that we desribe is universal in this sense.
It is universal in a seond way.
Allowing ontrats to speify ations that depend on other
ontrats means that ations might depend on whether other players' ontrats depend on the
way you make your ation depend on their ontrats, the way you make your ation depend
on how their ontrats depend on the way you make your ontrat depend on their ontrats,
and so on.
In simple prisoner's dilemma problems like the tax problem disussed above, this
problem is relatively straightforward sine 'dependene' simply means whether or not the other
player ooperates. However, in riher environments, 'dependene' is more subtle sine there are
many dierent ways that players an ondition their ations at eah round in the hierarhy of
dependenies desribed above. The method we desribe below provides a ompat way of dealing
with this.
Finally, the Bellman equation style representation of a reiproal ontrat illustrates that the
notion of reiproity depends on the ontrating environment beause the word 'ooperate' appears
in the denition of a reiproal ontrat. It isn't obvious how to extend the argument to problems
where a single 'ooperative' ation doesn't exist. The set of ontrats that we use, on the other
hand, is independent of the underlying game that is being played. Contrats need to map into
feasible ations, but the way that these ations depend on other ontrats doesn't depend on what
these ations are. Nor does it depend on whether or not players have private information. In this
sense, our ontrats are
universal
in the sense that they an be applied to any strategi situation.
DEFINABLE AND CONTRACTIBLE CONTRACTS
1.0.1.
How Denability Works.
5
Return again to the main purpose of denability. Instead of re-
ating speial terms like reiproal ontrat in an ad ho way to support ooperative outomes in
speial situations, we want to provide a ontrating environment in whih we an show that the
speial terms we need to write the ontrats that players need to enfore their ollusive agreement
will always exist within the language. We do it here to illustrate the method for the very simple
ase, then generalize the approah in the setions below.
Suppose there are
number of ations.
m
players in a normal form game in whih eah player has a ountable
Endow players with a formal language ontaining a ountable number of
words or haraters that they an use to write ontrats. Feasible ontrats are nite sequenes of
haraters in this formal language. As we mentioned above, the set nite subsets of a ountable
set is ountable, so there are bijetions mapping eah nite text into
alled the
Godel Coding
N.
. Provided the language inludes all the natural numbers and the usual
arithmeti operations, it is possible for players to write ontrats that are
NN −1
One suh a mapping is
denable
funtions from
into that player's ation spae. Sine denable funtions an be written as nite sequenes of
haraters in the language, they have Godel odes assoiated with them. Hene we ould interpret
the denable funtions as ontrats that make the players ation depend on the Godel ode of the
other player's ontrat.
To make the argument easier to relate to onventional ontrat theory, we assume below that
the ontrat spae for eah player is the set of denable funtions from
of the player's ation spaes
.
and onversely if the integer
NN −1
into the subsets
Every denable funtion an be assoiated with a unique integer,
n
is assoiated with a denable funtion, then it is assoiated with
a unique text. Now for eah array of funtions hosen by the players, ompute the Godel Code
of eah suh funtion. Fit the odes of the other players' strategies into eah player's strategy to
determine a unique subset of ations for every player. Then, players simultaneously take ations
from these subsets.
Our objetive is to try to haraterize the set of equilibria of this game. To see how it works,
we might as well restrit attention to a two player prisoner's dilemma. As we illustrated above, we
don't really need our formalism to understand this game. However, it provides a simple illustration
of the methods we use in the general ase. Call the players
the usual payo struture in whih
o if they both play
funtion from
N
to
C
D
strategy that hooses ation
[c]
D.
D
1
is
as the 'enoding' of
c1 ([c2 ])
and the ations
C
and
D
with
A strategy
c
for a player is a denable
no matter what the Godel ode of the other player's strategy.
c.
[c]
denote the Godel ode of the strategy
c
and
Sine the Godel oding is an injetion from the set of denable
strategies to the set of integers. For any pair of strategies
player
2,
One obvious equilibrium of this game ours when both players use a
Every denable funtion has a Godel ode. Let
refer to
and
is a dominant strategy and both players are stritly better
than they are if they both play
{C, D}.
1
c1
and
c2 ,
the ation (C or
D)
taken by
and similarly for player 2. Sine every pair of ations determines a payo, this
proedure assoiates a unique payo with every pair of strategies.
6
MICHAEL PETERS AND BALÁZS SZENTES
There are many things that aren't denable strategies that also have Godel odes. We want to
make use of some of these other things. In partiular, we want to use denable strategies with
variables.
free
For example, there is a sublass of denable strategies for player 1 dened parametrially
by
γ x (n) =

C
n = x,
D
otherwise.
free variable x
This is simply a denable strategy with a
, where
x
is the target ode of the other
player's strategy that will trigger the ooperative ation. Denable strategies with free variables
are also denable, and so they too have Godel odes. The strategy with free variable that we want
is a slight modiation of the one above, in partiular
cx (n) =
(1.1)
The mapping
< x >(x) is
a text with one free variable, then
n.
Hene, if
n
i
h
n = hxi(x) ,
D
otherwise.
the omposition of two funtions. First, the funtion
operation to the Godel oding. That is,
to be

C
φ
(n)
<n>
is the text whose Godel ode is
hxi
n.
is the inverse
Seond, if
φ
is
is the same text where the value of the free variable is set
is a Godel ode of a denable strategy with one free variable, then
is itself a denable strategy (without a free variable).
h
i
(n)
hni
this denable strategy happens to be. Notie that in this ase,
< n >(n)
is just the Godel ode of whatever
[hxi (x)]
won't be equal to
x sine a
denable strategy must have a dierent Godel ode from a denable strategy with one free variable
beause of the fat that the Godel oding is injetive.
We want to dene a strategy by xing a value for
are interested in is
[cx ].
Sine
[cx ]
x
in (1.1). In partiular, the value of
x
we
is the Godel ode of a strategy with a free variable, the right
hand side of (1.1) requires that we deode
[cx ]
Putting all this together gives
c[cx ] (n) =
So
c[cx ] ([c2 ]) =
to get

C
cx ,
then x
x
at
[cx ]
to get the ontrat
c[cx ] .
n = c[cx ]
D
otherwise

C
[c2 ] = c[cx ]
D
otherwise
is a the 'reiproal' or self-referential ontrat mentioned above. Now we simply need to verify
what happens when both players use strategy
If player 2 uses strategy
by player 1.
c[cx ] ,
then
c[cx ] .
[c2 ] = c[cx ]
, whih evidently triggers the ooperative ation
The same argument applies for player 2.
denable strategy
c′
Player 2 an deviate to any alternative
that she likes. Sine every denable strategy has a Godel ode, the reation of
player 1, and onsequently both players payos are well dened. As the Godel oding is injetive,
DEFINABLE AND CONTRACTIBLE CONTRACTS
c′ 6= c[cx ]
implies the Godel ode of
respond by swithing from
C
to
c′
is not equal to
D.
c[cx ]
7
, and the deviation by 2 indues 1 to
Notie that this argument makes use of an enoding of the strategy with free variable
isn't a denable strategy.
cx ,
whih
One might have expeted the target ode number to be assoiated
with a strategy instead of a strategy with a free variable. For example, it seems that to enfore
ooperation there needs to be a denable strategy
c∗ =
Of ourse, for arbitrary
n∗

C
c∗
with enoding
[c∗ ] = n∗
suh that
[c2 ] = n∗
D
otherwise
[cn∗ ] = n∗ .
it will be false that
This leads to a xed point problem
that, in fat, does not have a solution in general. More generally, one ould try to onstrut a
self-referential ontrat by nding a xed point of the the following problem. For eah
cn ([c2 ]) =
g
where

C
D
if
[c2 ] = g (n) ,
n∗
suh that
[cn∗ ] = g (n∗ ),
a self-referential ontrat. Indeed, what we did above is that we hose
showed that
n = [cx ]
reiproal ontrat
≡
and its reursive ounterpart
C
=
don't
c[cx ]
to be
cn∗
is obviously
[< n >(n) ]
and

exempt
don't
cx
works, reall the reiproal tax agreement
other State oers reiproal ontrat
exempt
otherwise
if other State exempts any State who exempts any State who exempts. . .
otherwise
The 'reiproal ontrat' is
g (n)
then
is a orresponding xed point.
To see how the strategy with free variable

exempt
onsider
otherwise,
is a denable funtion. If there exists an
∗
n,
c[cx ]
and the statement other state oers reiproal ontrat is
[c2 ] =
.
State
A
wants to exempt any state whose law fullls a ondition. For example, if the ondition
it is looking for is that the other state simply exempts State S, then it would ompute the Godel
ode
n0 = [C∀n]
then use the strategy
cn0 =

C
D
[c2 ] = n0
otherwise
If it does that, then it an't be an equilibrium as explained above.
So what it needs to do is
to exempt any State whose law fullls a ondition that exempts any state whose law fullls a
ondition. For example, if it wanted to exempt State
A
if and only if State
and so on.
A
B
unonditionally exempts state
if and only if State
B,
B 's
law exempts state
then it would adopt the strategy
c[cn ] ,
0
8
MICHAEL PETERS AND BALÁZS SZENTES
cx omes
i
h
(x)
,
n = hxi
This is where the partiular struture of the ontrat
cx (n) =

C
D
into play. Reall that
otherwise.
It speies exemption if and only if a ondition is fullled, but it doesn't seem to speify what the
ondition is. However, it does require that whatever the ondition
x
is, if
x
in turn depends on
a ondition, then the ondition that it depends on must be the same as the ondition itself. To
see if
x
depends on a ondition, we rst deode it and nd the statement
hxi
that the integer
orresponds to. Then if it depends on some ondition, we require that that ondition be
whih is the meaning of
exempts state
B 's
B
law must be
(x)
hxi
. So now we an do the innite regress. State
if and only if the Godel ode of State
c[cx ] ,
or that
i.e., the same ondition that
B
A
exempt
A
B 's
law is
c[cx ] .
A
itself,
adopts a law that
This means that state
if and only if the Godel ode of State
requires.
x
x
A's
law is
c[cx ]
,
Every olletion of denable ontrats uniquely determines a set of ommitments for eah of the
players. Any sensible desription of the set of feasible ontrats should unambiguously determine
players' ommitments in this way. We aomplish this by making the ontrats arithmeti. The
set of denable funtions is the largest set of arithmeti funtions that an be desribed using
a nite text in a rst order language.
In this sense, the lass of ontrats that we desribe is
universal in that any 'sensible' model of ontrats on ontrats should involve a ontrat spae
that is embedded in the one we desribe.
2.
Literature
As we mentioned in the introdution, our paper is not the rst to show how ontratual devies
an be used to support ooperative play.
Muh of the literature in this area follows an idea
developed in [5℄ in whih ations are delegated to an agent who is given the appropriate inentives
to arry out ations that might not otherwise be part of a non-ooperative equilibrium. This idea
was developed by [8℄ who used it to prove a 'folk theorem' for a very speialized environment.
The idea that the agent might be used to report deviations, thereby allowing prinipals to ommit
themselves to punish a deviator, is developed in [4℄.
This idea provides the basis for the menu
theorems in ommon ageny, like [9℄, [10℄ and [6℄ whih illustrate how ooperative outomes an
be supported by having the agent trigger punishments. Reently [14℄ provides a very general folk
theorem for multiple ageny games in whih prinipals an ommit to follow the reommendations
of (potentially interested) agents.
Though this literature shows how ontratual devies an be used to support ooperative behavior, it relies on the delegation of deision making power to an agent. In this paper, there is
no agent, beause ontrats depend diretly on one another. This approah is losely related to
ideas in the omputer siene literature. One paper that uses this approah is [13℄. He has players
writing programs that determine their ations. Using an idea due to von Neumann, he allows these
DEFINABLE AND CONTRACTIBLE CONTRACTS
9
programs to use other programs as data, whih has the eet of making the output of eah player's
program depend on the other players' programs. The result extends the reiproal ontrat idea
presented above to a more general
n
player game. We have illustrated the basi priniple with our
'ross-referential' example above. To support any array of ations, Tennenholz eetively writes
out expliitly a sequene of programming statements resembling the verbal statements we made
above. These dene the keywords that are needed to support any array of ations in whih eah
player's payo is at least his or her minmax value.
The paper by [7℄ shows how to extend the set of supportable alloations in two player games.
They onstrut a set of ommitment devies whih an be used to support
orrelated
strategies in
whih all players payos exeed their minmax payos. Speially, in some games their devies
support outomes in whih all players reeive payos that exeed their best payos with Tennenholz's programs. This is aomplished by onstruting ommitment devies that allow players to
orrelated their ations while using independent randomizing devies.
On the most basi level, our paper diers sine we are interested in problems with inomplete
information. However, the more important dierene is that our approah is dedutive rather than
onstrutive. We x a set of ommitment devies, then use this same set of devies to support
individually rational outomes in all nite games. The advantage of this is that we are able to
give a omplete haraterization of the set of all supportable alloations. Tennenholz theorem does
not rule out, for example, the possibility that there might be program equilibrium in whih some
players reeive less than their minmax payo. With omplete information, this possibility is not
ritial. For example, [7℄ rule it out by allowing players to reserve the right to pik their ations
in an unonstrained way ex post. They interpret this as giving players the right not to partiipate
in the ontrating proess. With inomplete information, there is no simple analog to the minmax
payo, so there is no simple trik like non-partiipation that an be used to show whih alloations
annot
be supported as equilibria. By providing a more omplete desription of the set of feasible
ontrats we are able to overome this diulty.
It is preisely the ability to pin down alloations that annot be supported as equilibrium that
is ritial to our objetive of showing the limits to whih ontrats an be used to deentralize
the mehanism designer's problem. We use our haraterization to onstrut examples of alloation rules that an be supported by a mehanism designer, but annot be supported as ontrat
equilibrium.
We emphasize that the ontribution here is not intended to be a ontribution to the omputer
siene literature.
In fat, we view the paper as a very traditional ontrating model in whih
players have aess to a legal system whih an be used to provide redress when ontrats are
not arried out. Yet redress is all we want. Our purpose is to dene a ontrating language suh
that players an write any ontrat that they like in this language.
One all the players have
written their ontrats, they should be able to dedue on their own what ations they need to
take in order to fulll their ontrats. With omplete information, it isn't ompletely surprising
10
MICHAEL PETERS AND BALÁZS SZENTES
that many alloations an be supported in suh an environment. It is in inomplete information
environments where the set of supportable alloations has not been haraterized, and this is how
we view our ontribution here.
Finally, the use the Godel oding is simply for onveniene. One one sees that the olletion
of all nite texts onstitutes a ountable set, any denable bijetion from nite texts to integers
an be used to do the analysis we do. Any denable bijetion an be expliitly written into the
ontrats we allow, so that a judge (or player for that matter) who doesn't know what it is an
expliitly alulate it.
3.
The Language and the Gödel Coding
We onsider a formal language, whih is suiently rih to allow its user to state propositions
in arithmeti. Furthermore, the set of statements in this language is losed under the nite appliations of the Boolean operations:
q, ∨,
and
∧.
This implies that one an express, for example,
the following statement:
∀n, x, y, z {[(n ≥ 3) ∨ (x 6= 0) ∨ (y 6= 0) ∨ (z 6= 0)] → (xn + y n 6= z n )]} .
In addition, one an also express statements in the language that involve any nite number of free
variables. For example, x is a prime number is a statement in the language. The symbol
a free variable in the statement.
x
< 4.
Let
L
is
Another example for a prediate that has one free variable is
One an substitute any integer into
x = 0, 1, 2, 3
partiular one is true if
x
x
and then the prediate is either true or false. This
and false otherwise.
be the set of all formulas of the formal language. Eah of its element is a nite string
of symbols. It is well known that one an onstrut a one-to-one funtion
value of this funtion at
ϕ ∈ L,
and all it the Gödel Code of the text
L → N.
Let
[ϕ]
be the
ϕ.
In what follows, we dene a lass of funtions whih an be represented represented by nitely
many haraters in our formal language.
Denition 1.
φ
in
k+1
The funtion
f : Nk → 2 N
free variables suh that
In the denition, the mapping
singleton for all
we refer to
[φ]
Example.
n∈N
k
, then
f
is said to be
b ∈ f (a1 , ..., ak )
f
denable
if and only if
is a orrespondene from
is a funtion. If the funtion
as the Godel enoding of
f.
(
f
φ (a1 , ..., ak , b)
Nk
to
N.
is true.
Of ourse, if
f (n)
is denable by the prediate
φ
is a
then
We illustrate the previous denition with an example.
Consider the following funtion dened on
f (a) =
if there exists a rst-order prediate
N:
0
if
a
is an even number,
1
if
a
is an odd number.
We show that this funtion is denable by onstruting the orresponding prediate
φ (x, y) ≡ {{y = 1} ∧ {y = 0}} ∨ {∃z : 2z = y + x} .
φ.
DEFINABLE AND CONTRACTIBLE CONTRACTS
φ indeed has two free variables.
Notie that
φ
front of it.) The rst part of
that
φ (a, b)
f (a) = 0
or one.) If
beause
b=0
a+b
φ (a, 0)
then
Suppose there are
payo of Player
is not free beause there is a quantier
is either one or zero. The seond part says that
φ (a, 0)
if and only if
is true. To see this, rst notie
φ
(This is beause the rst part of
is indeed true. If
b = 1,
x+y
b
requires
then the seond part of
φ
to be zero
beomes false
Complete information Contrating Game
m
players. Player
i is ui (a1 , . . . , am ).
i if he takes ation ai
ontrat,
submits a
z
is an odd number.
4.
to player
b∈
/ {0, 1}.
is false whenever
y
states that
is divisible by two. Notie that
(The variable
11
i
Ai .
has a nite ation spae
Let
We use the onventional notation that
while the other players take ation
N
whih is a denable orrespondene from
m
a−i .
to
A
denote
×m
i=1 Ai .
The
ui (ai , a−i ) is the payo
Eah player simultaneously
N
2 ,
where `denable' is to be
understood in the sense of Denition 1. At stage two, players take ations simultaneously from
subsets of their ations spaes. These subsets are determined by the rst-stage ontrats. If at
stage one player
two if
j
submitted ontrat cj (j
[ai ] ∈ ci ([c1 ] , ..., [cm ]).
= 1, ..., m), then player i an only take ation ai at stage
We restrit attention to pure-strategy subgame perfet equilibria of
this game.
The pure strategy minmax value for player
ui =
Let
aj
be any one of the ations that
for player i, suh that,
i
min
is
max ui (ai , a−i ) ,
a−i ∈A−i ai ∈Ai
j
uses to attain his minmax payo. Let us x an ation
aj1 , ..., ajm ∈ arg min uj aj , a−j .
aji i
a−i
j
That is, ai is the ation that player
all
i
uses to punish player
j.
For onveniene, dene
ajj = aj
for
j ∈ {1, ..., m}.
Theorem 1.
The ation prole
a∗ = (a∗1 , ..., a∗m ) ∈ A
outome in the ontrating game if and only if
is supportable as a pure-strategy SPNE
∗
ui (a ) ≥ ui
for eah i.
Before we proeed with the proof of the theorem, we reall two piees notations from the
introdution.
[< n >] = n.
First, if
n ∈ N
Seond, for any text
stands for a free variable in
is
x1 < x2 , n1 = 1,
variable:
< xi >
and
ϕ
then
n2 = 2
(x1 ,...,xk )
< n >
then
ϕ,
xi
then
, where
let
ϕ
denotes the text whose Gödel ode is
(n1 ,...,nk )
ϕ
i ≤ k.
is
1 < 2.
ni
in
ϕ(n1 ,...,nk )
is dened for all
Mathematial Logi, that if
ϕ
ϕ
for
i = 1, ..., n.
For example, if
Consider now the following text in
One an evaluate this statement at any
vetor of integers. Sine the Godel oding was a bijetion
addition,
and
That is,
denote the statement where if the letter
is evaluated at
(n1 ,n2 )
n.
(n1 , ..., nk ).
< ni >
k
xi
ϕ
free
k -dimensional
is a text for eah
n i ∈ N.
In
In addition, it is a well-known result in
f (n1 , ..., nk ) = < ni >(n1 ,...,nk ) ,
then
f
is a denable funtion.
12
Proof.
MICHAEL PETERS AND BALÁZS SZENTES
First, we prove the only if part. Fix an equilibrium in the ontrating game. Let
j (j = 1, ..., m)
the equilibrium ontrat of player
i
Notie, that player
he an oer
c:N
m
oering
is,
ej
A
3
c instead of ci .
suh that
c (n1 , ..., nm ) = N
ui < ui ,
We show that if
Let
e
cj = cj
if
is the ation spae of player
Also notie that
ui
denote player
i's
denote
equilibrium payo.
an always oer a ontrat that does not restrit his ation spae. That is,
→ 2N ,
obviously denable.
and let
cj
fi = Ai .
A
j
player
j 6= i and e
ci = c.
for all
i
(n1 , ..., nm ) ∈ Nm .
The ontrat
i
ej = {aj : [aj ] ∈ e
A
cj ([e
c1 ] , ..., [e
cm ])}.
min
The inequality follows from
max ui (ai , a−i ) ≥
ej ⊆ Aj
A
for all
minmax value by oering the ontrat
j.
That
(e
c1 , ..., e
cm ).
in any pure strategy equilibrium of this subgame
is weakly larger than
e−i ai ∈Ai
a−i ∈A
is
an protably deviate at the rst stage by
Let
in the subgame generated by the ontrat prole
The payo of player
c
min
max ui (ai , a−i ) .
a−i ∈A−i ai ∈Ai
Therefore, player
i
an always ahieve his pure
c.
For the if part, onsider the following ontrat of Player i,
m
cixi ,x−i ,
in
m
free variables:
cix1 ,...,xm ([cj ])j=1 =

 [a∗ ]
h ii
 aji
(4.1)
| k : < xk >(x1 ,...,xm ) 6= [ck ] | =
6 1,
(x1 ,...,xn )
if
k : < xk >
6= [ck ] = {j}
if
The expression (4.1) is not a ontrat, but rather a ontrat with free variables.
Eah suh
o
n
i
cγ ,...,γm
The funtions
have no free
expression has a Godel ode, so let
i
om
n 1
i
variables, so they onstitute a set of ontrats. We will now show that cγ ,...,γ
onstitutes
1
m i=1
1
m
. First observe what
an equilibrium prole of ontrats whih support the outome ak , . . . , ak
1
m
γ i = cix1 ,...,xm .
happens when all players use ontrat
Player
player
i
ciγ 1 ,...,γm .

 [a∗ ]
m
h ii
ciγ 1 ,...,γm ([cj ])j=1 =
 aji
if
if
Notie that
| k : < γ k >(γ 1 ,...,γm ) 6= [ck ] | =
6 1,
(γ 1 ,...,γ m )
6= [ck ] = {j} .
k : < γk >
needs to hek whether the Godel ode of
k 's ontrat, ck .
γk
is equal to the Godel ode of
is the Godel ode of the ontrat with free variable
ckx1 ,...,xm .
γ 1 , ..., γ m
k
(whih gives the ontrat cγ ,...,γ ), then evaluate its Godel ode. This is what is to be ompared
1
m
Player
i's
The integer
< γ k >(γ 1 ,...,γm )
ontrat says to take this ontrat with free variable, x the free variables at
with the Godel ode of the ontrat oered by
k.
Of ourse, these are the same in equilibrium
k
beause ck = cγ ,...,γ . Sine this is the ase for all m − 1 of the other players, player i ends up
1
m
∗
taking ation ai . So these ontrats support the outome we want if everyone uses them.
3
For example, the prediate
{x1 = x1 } ∧ ... ∧ {xm = xm } ∧ {y = y}
denes c. That is, for all y ∈ N the prediate is true no matter how the free variables are evaluated.
DEFINABLE AND CONTRACTIBLE CONTRACTS
Nm
13
2N . o
However, any suh ontrat
n
j
will have a dierent Godel ode, and so will indue the punishment ai
from the other players.
i6=j
n o
j
Reall that ai
is the ation prole that players other than player j use to minmax player j .
Player
Sine
j
an deviate to any denable ontrat mapping
uj (a) ≥ uj
into
i6=j
any deviation will be unprotable.
One might argue that restriting the spae of ontrats to be denable funtions of Godel odes
is both arbitrary and unnatural. Indeed, there is no reason for a judge to interpret a ontrat as
a desription of a mapping from the Godel odes of the ontrats oered by the other players to
the ations spae of the player. For that matter, the judge might not even know about the Godel
oding. It is important to note that the salient feature of denable ontrats is that they an be
written as texts that use a nite number of words in a formal language. The set of nite texts
seems a very natural desription of the set of feasible ontrats. In fat, from this perspetive it
seems that
any
reasonable desription of the set of feasible ontrats should allow any suh text.
The ompliation with suh a broad desription of the set of ontrats is that to properly dene
a game, one must fully desribe the mappings from proles of texts into payos.
Many texts
will be omplete nonsense and some modelling deision has to be taken about how these would
translate into ations and payos. The ontrats that we speify above are denable texts that
have two advantages in this regard. First, sine every nite text has a Godel ode, they tie down
the ation of the player who oers suh a ontrat even if the other players in the game oer
ontrats involving texts that make no eonomi sense. Furthermore, if all players oer ontrats
from the set we speify, an outome for every player is uniquely determined.
Finally, sine the Godel oding itself is denable, the oding an be embedded diretly into the
ontrat. So players don't need to agree to use the Godel ode of other ontrats. They an use
the Godel ode unilaterally, and the impliations of the ontrat will be understood by the others
provide they agree on the underlying language in whih ontrats are written.
Generalizations.
Everything about this theorem involves pure strategies. This imposes limits
on its appliation. Next, we disuss how to extend our result to the ase when players an mix
over their restrited ation spae at the seond stage of the game but annot randomize over the
ontrats they oer at the rst stage.
Allowing suh mixing expands the set of payo proles
that an be supported by equilibria for two reasons. First, sine players an randomize ertain
onvex ombinations of payo proles an now be supported. Seond, players an use mixing when
punishing a deviator, and hene the minmax value of the players will be smaller.
Formally, for all
S = ×i Si , Si ⊂ Ai ,
and the payo funtion of player
i
dene a game,
is the restrition of
GS ,
ui
where the ation spae of player
on
S.
Let
E (S)
∗
equilibria in GS . Dene the minmax value of player i, ui , as
u∗i =
min
max
min
S−i ⊂A−i Si ⊂Ai σ∈E(S−i ×A)
S−i =×j6=i Sj
Z
ui (a) dσ (a) .
i
is
Si ,
denote the set of mixed
14
MICHAEL PETERS AND BALÁZS SZENTES
The idea is that in the ontrating game, players an restrit their ation spaes arbitrarily, hene,
when they punish player
i
they an hoose
S−i
arbitrarily. On the other hand, their seond-stage
ations must be best responses, and that is why we have to onsider equilibrium payos in the
restrited game. An argument idential to the proof of Theorem 1 shows that the random alloation
σ ∈ ∆ (A)
an be supported as an equilibrium if
∃Si ⊂ Ai for all i, suh that σ ∈ E (×i Si ),
R
ui (a) dσ (a) ≥ u∗i for all i.
(ii)
(i)
and
What happens if players are allowed to randomize over the ontrats they oer? It is possible
to show that part (i) an be ompletely relaxed. That is, the distribution over the outomes does
not have to be an equilibrium in
GS ,
and it does not even have to be generated by independent
randomizations of the players. The onstrution of mixed equilibria in our ontrating game that
supports orrelated outomes is entirely based on Kalai et.al. (2008). The authors onsider twoperson games where players submit ommitment devies instead of taking ations. A devie then
determines the ation of the player as funtion of the other devie. The authors onstrut a set
of devies suh that any individually rational orrelated outome an be implemented as a mixed
equilibrium in the game. That is, although the players mix independently over their devies, the
distribution over the ations proles will be orrelated. It is easy to show that these results extend
to
n-person omplete information games, and in addition,
the the equilibrium ommitment devies
onstruted by Kalai et.al. (2008) are denable funtions as long as the probabilities involved in
eah mixing are all rational numbers.
Theorem 2.
Suppose that
σ ∈ ∆ (A),
and
σ (a) ∈ Q
for all
a ∈ A.
The distribution
σ
an
be supported as a mixed-strategy equilibrium outome in the ontrating game if and only if
R
ui (a) dσ (a) ≥ u∗i
for all
i ∈ {1, ..., m}.
Another question is why we use denable funtions as opposed to programs or Turing mahines.
One might want to require that the ontrats must be omputable and assume that the set of
available ontrats is the set (or a subset) of Turing mahines.
(i = 1, 2) hooses mahine τ i , then τ i
runs on the desription of
In suh a model, if player
i
τ j , and the output will be a subset
of the ation spae of player i. It is well-known, that one an onstrut self- and ross-referential
4
ontrats (mahines) in this spae too.
In fat, this onstrution is essentially idential to our
onstrution of ross-referential denable funtions. Most importantly, the equilibrium ontrats
we onstrut to support individually rational alloations are, in fat, reursive funtions, and hene
they are omputable by Turing mahines. Therefore, if the reader insists on omputability, he an
restrit attention to the spae of Turing mahines.
There are, however, several advantages of our approah over modelling ontrats with Turing
mahines. First, Turing mahines do not always halt, and therefore, it is not lear how one an
4
Suh mahines were onstruted even in the ontext of Game Theory, see Anderlini 1990 and Canning 1992.
DEFINABLE AND CONTRACTIBLE CONTRACTS
15
dene the restrition on the ation spae of a player, if his mahine does not halt.
A way to
handle the halting problem is to restrit the spae of Turing mahines to be the set of mahines
that always halts. We nd suh restritions arbitrary. Instead of restriting the spae of reursive
funtions, we expanded it to be the set of denable funtions and avoided the halting problem that
way. Seond, another problem with Turing mahines is that they an only ondition on the atual
desription of the mahines submitted by the other players but annot ondition on the funtions
what the mahines ompute. Take the example of the prisoner dilemma. It is possible to onstrut
a Turing mahine,
τ,
suh that
τ ([τ 2 ]) =
(
C
D
if
[τ 2 ] = [τ ]
otherwise.
The problem is that if player 2 submits a mahine, say
with
τ,
τ ′,
whih is omputationally equivalent
but has a dierent desription, then player 1 would defet.
In fat, it is not possible
to onstrut a mahine whih does not suer from this problem. We avoid suh problems with
denable funtions. Indeed, it is possible to express ontrats that do not ondition on the atual
way the other ontrat is written, but on the funtion itself that the other ontrat desribes.
Consider
c1 ([c2 ]) =
The ontrat
cγ
(
c∗2 ⇔ c2 ,
C
if
D
otherwise.
is obviously denable, but does not ondition on the atual form of
c2 .
As long
∗
as c2 represents the same funtion as c2 , ooperation is presribed.
5.
Contrating in a Bayesian Environment
In the previous setion, we showed how ontratible ontrats an be used to support any
alloation for whih every player's payo is at least his minmax value. Assuming non-partiipation
is always an option, this is the set of alloations that is supportable by a entralized mehanism
designer. In this sense, ontratible ontrats ompletely deentralize the alloation problem. In
this setion, we show that the same result is not true in the Bayesian ase. We do this by proving a
theorem that ompletely haraterizes the set of alloation rules that an be supported as Bayesian
equilibrium in the ontrating game. We then onstrut alloations that an be supported by a
entralized mehanism designer, but whih annot be supported as equilibria.
We also show, however, ontratible ontrats make it possible for one player's ation in a
ontrat equilibrium to depend on another player's type. The reason is that ontrats expliitly
ondition ations on other player's ontrats, whih, along the equilibrium path, an depend on
their types. We exploit the reiproal ontrating idea desribed above to enfore type ontingent
agreements. The idea is that a ontrat will speify a number of 'target' Godel odes, one for eah
of the ontrats the other player's dierent types are supposed to oer along the equilibrium path.
16
MICHAEL PETERS AND BALÁZS SZENTES
As long as other players oer a ontrat whose ode is equal to one of these targets, the ontrat
responds with a 'ooperative' ation. If any of the others deviate, the ontrat will respond with
some kind of punishment.
When information is omplete, a punishment is simply an array of ommitments that nondeviators make.
These ommitments are suh that they make all deviations unprotable.
We
want to extend this idea to the Bayesian ase. There are a number of diulties assoiated with
this. First, ontrat oers depend on player types, so they reveal information. Potential 'deviators'
an ondition their play on non-deviators' ontrats, so they an ondition their ommitments on
the non-deviators' type. Players may not want to reveal their type information at the ontrating
stage for this reason.
Nonetheless, they will want at on this type information ex post.
So
ontrats will typially bind players to subsets of their ations, leaving some disretion for them
to vary their ations with their types ex post.
So it is important to think of ontrat oers as
ommitment orrespondenes, rather than simple ommitments to ations as would sue with
omplete information.
Seondly, non-deviators have the ability to respond to deviations ontratually. They an make
their punishments more severe by exploiting residual unertainty that the deviator has about
their type ex post. They would do this, again, by ommitting to a subset of ations instead of a
single ation. The punishment for a deviation onsists of two parts for this reason: a punishment
orrespondene, that restrits the non-deviator's ex post hoie to a subset of his ations, and a
seond stage strategy that depends on the non-deviator's type (onstrained by the information the
non-deviator reveals about his type with his on path ontrat).
Perhaps the most omplex part of ontrating equilibrium is that non-deviators' responses depend on the deviator's ontrat.
sophistiated way.
As suh they ould, in priniple, depend on the deviation in
The simple logi that we developed for the reiproal ontrating argument
above relied on the idea that non-deviators ould
punish
deviators by minmaxing them.
The
minmax ation doesn't depend on exatly how the non-ooperative player hooses to be nonooperative. With inomplete information, there is no natural analog for the minmax punishment.
The ability to ontrat on other ontrats suggests the possibility that non-deviators ould reat to
deviations in a way that depends on exatly what the deviation is thereby holding on path payos
to something below the orresponding min max payo.
A remarkable property of our theorem is to show the sense in whih punishments an be understood to be invariant to the manner in whih a player hooses be non-ooperative. We show
that any ontrat equilibrium an be supported by having non-deviators respond to a deviation
with a ontratual ommitment that is independent of what the deviator hooses to do. Ex post
hoies that non-deviators make from their ontratual ommitment orrespondenes will typially
depend on the deviation.
trats.
However, these ex post hoies aren't part of the non-deviator's on-
This result allows us to extend the reiproal ontrating idea to Bayesian games.
If a
player oers a ontrat that is onsistent with equilibrium play by one of his types, then the other
DEFINABLE AND CONTRACTIBLE CONTRACTS
17
players respond ooperatively. Otherwise, there is a single punishment orrespondene that the
non-deviators impose, just as in the omplete information ase.
This property of ontrat equilibrium is a onsequene of restriting players to denable ontrats. It is the part of our theorem that allows us to show the limits of ontratible ontrats,
sine we an show the kinds of alloations that an't be supported as equilibrium.
This is the
part of our theorem that we use to provide examples of alloations supportable by a mehanism
designer but not by ontratable ontrats. This is the advantage we derive from speifying the
ontrat spae very preisely. An abstrat ommitment spae suh as the one provided in [7℄, or a
spae of ontrats that is onstruted to have desirable properties, suh as programs in [13℄, an be
used to show that a large set of alloations an be supported as equilibrium alloations. However,
in the Bayesian ase, a omplete haraterization requires a demonstration that ertain alloations
annot be supported.
To provide this, a omplete desription of the set of feasible ontrats is
required. Our Lemma illustrates that denability provides just suh a omplete desription.
The model is the same as in the previous setion, with the addition of player types. There are
m
players. Player i's ations spae is a nite set denoted by
from a nite set
T
i
Ai .
Eah player
i
has a type ti drawn
. The joint distribution types is ommon knowledge. The payo of player
ui (ai , a−i , t) where t ∈ T1 × · · · × Tm .
Notie that a strategy rule for player
i in the
i
is
Bayesian game
|Ti |
.
the players might otherwise be involved in is an element of Ai
Our haraterization of the set of alloations that an be supported as ontrat equilibrium
hinges on the information that equilibrium play reveals about players' types. Our argument refers
repeatedly to the information that is revealed through equilibrium play. Most things that happen
o the equilibrium path depend on this information.
use the
information partition
orrespondene
type
ti .
τ i : Ti → 2
Ti
A natural way to inorporate this is to
indued by these ontrats.
Fix an equilibrium, and dene the
to mean the set of types of player
One other players see the ontrat oered by player
believe that
i's
type lies in the set
τ i (ti ).
i
i
who oer the same ontrat as
of type
The orrespondene
τi
ti ,
is an
they should ommonly
information partition.
Similarly, the orrespondene
τ −i (t−i ) =
Y
τ j (tj )
j6=i
desribes the information available to player
i
about the types of the other players.
Eah ontrat speies a set of ations from whih players subsequently hoose. In this sense,
equilibrium ontrats support a ommitment
orrespondene
for eah player. As ontrats depend
on other ontrats, whih in turn depend on other players' types, this ommitment orrespondene
an be written as a mapping
ri : T → 2Ai .
Sine the set from whih
i
hooses his ation an only
depend on some other player's type to the extent that the other player's ontrat varies with his
type,
ri
should be measurable with respet to the information partition
τ −i .
Contrats speify sets of feasible ations. Ultimately, payos are determined by players' hoies
from these sets in the nal stage of the game.
Let
si : T → Ai
denote the outome funtion
18
MICHAEL PETERS AND BALÁZS SZENTES
assoiated with the seond stage strategies.
that
si (t)
lies in the set
depend on
t
instead of
ri (t)
ti .
5
These outome funtions must have the property
for eah t. It might seem strange that this outome funtion should
The reason that player
other players is twofold. First, player
i's
equilibrium ations depend on the types of
i gets to see the ontrats oered by eah of the other players.
His beliefs vary as the other players' ontrats vary, so his ations in the seond stage will vary
with the other players types. Seondly, his own ommitments depend on the ontrats, and thus
the types of the other players. Evidently, player
i
only observes types imperfetly by observing
the ontrats that are oered. This is aptured simply by observing that this indued outome
funtion must be measurable with respet to the information partition
τ −i .
Eah o equilibrium ontrat oered by a deviator speies a ommitment for eah array of
ontrats oered by the other players. Sine the ontrats the other players oer depend on their
types, a deviation implies a ommitment orrespondene
fi : T−i → 2Ai .
Sine these types are
revealed only through the ontrats that the others oer, this orrespondene should be measurable
with respet to the information partition
τ −i
that aptures this information. Let
all ommitment orrespondenes available to the deviator, i.e.,
mappings from
T−i
into
2
Ai
Fi
is the set of all
Fi
be the set of
τ −i
measurable
.
In a ontrat equilibrium, a deviation leads to two sorts of 'punishments'. First, sine the other
players' ontrats speially ondition on the ontrat oered by the deviator, the non-deviators
will hange their ommitments. As mentioned above, we are going to show that the punishment
orrespondene assoiated with this hange in ommitments an be taken to be independent of the
deviation
fi .
However it is possible that the way that the non-deviators hoose from sets to whih
they have ommitted themselves will depend on
i
p =
j
Q
fi .
i deviates as pij : T−i → 2Ai and
i
j6=i pj . In a ontrat equilibrium, this punishment is the onsequene of the ontrat that
Write the 'punishment' that player
j
imposes when player
has written, so the punishment an only vary with
j 's
type to the extent that
j 's
ontrat does.
As a onsequene, this punishment will be measurable with respet to the information partition
τ −i .
Finally, we need to desribe the non-deviators' behavior in the ex post stage, let
T−i → Aj .
equilibrium
The non-deviator
j
sij : Fi ×
an no longer ondition his behavior on information revealed by i's
behavior. However, he does observe i's ommitment orrespondene in the sense that
he observes the deviator's ontrat. He also observes the on equilibrium ontrats of the others.
This is aptured, as always, by requiring that his behavior be measurable with respet to
5
τ −ij .
The
To simplify the argument slightly, we fous on pure strategy outomes here. It is ompletely trivial to extend
this argument to outomes that involve randomization at the seond stage by having the outome funtions be
mappings from T into △ (Ai ), restriting the supports of these mappings to lie in ri (t), then letting ui (s (t) , t) be
the expeted utility assoiated with the randomization. With this notation, the inequalities that haraterize the
equilibrium remain unhanged.
DEFINABLE AND CONTRACTIBLE CONTRACTS
ation
sij (fi , t−i ) should be ontained in pij (t−i ) for every t−i ∈ T−i
be the outome funtion assoiated with a deviation. for eah
and
19
fi ∈ Fi .6
Let
si =
i = 1, . . . m,
Q
j6=i
sij
The theorem is based on a pair of inequalities that are based on the objets dened above. The
rst aptures the idea that no player wishes to mimi the equilibrium behavior of another of his
own types. For eah
i = 1, . . . m,
and eah
ti
and
t′i
Et−i (ui (s (t) , t) : ti )
(5.1)
≥ Et−i
max
ai ∈ri (t′i ,t−i )
Et′−i
The ommitment orrespondene
ri
ui ai , s−i t′i , t′−i
,
ti , t′−i
:
t′−i
∈ τ −i (t−i ) : ti
!
results in a olletion of ations from whih player
.
i is ommit-
ted to hoose in the seond stage. This hoie set an depend on the types of the others beause
max
their ontrats do. The
operator on the right hand side requires the player to hoose a best
reply from this set given posterior beliefs.
Taken together, these onstraints for all the players
requires that play in the seond stage onstitutes a Bayesian equilibrium of the game in whih
eah player hooses an ation from the set of ations to whih he is ommitted, given posterior
beliefs about players' types.
To deal with deviations at the ontrating stage of the game, we require that for eah ti
∈T
Et−i (ui (s (t) , t) : ti ) ≥
(5.2)
max Et−i
fi ∈Fi
max Et′−i ui a, si fi , t′−i , ti , t′−i , t : t′−i ∈ τ −i (t−i ) : ti .
a∈fi (t−i )
A deviation implies a ommitment orrespondene
fi .
The inequality says that even if the deviator
hooses a best reply from the set of ations to whih he is ommitted, he annot gain by deviating.
So far we have imposed no restritions on seond stage punishment behavior. In appliations it
is natural to want behavior in the seond stage to be onsistent with some renement like
perfet
Bayesian equilibrium. Our theorem works with most standard renements but does not depend
on them. To illustrate, let
T̃−i
be a subset of
T−i .
Ã1 , . . . , Ãm
be a olletion of subsets of the players ation sets and let
while it is ommon belief that the non-deviators' types lie in
the subset of ations in
Ã−i
ned) Bayesian equilibrium,
have
i
h
Ri Ã1 , . . . , Ãm , T̃−i
6
Ri
has deviated from
the seond stage,
T̃−i .
Let
Ri Ã1 , . . . , Ãm , T̃−i
i
be
that are onsistent with some renement. For example in a (unrei
h
Ri Ã1 , . . . , Ãm , T̃−i = Ã−i . A slightly stronger renement might
ruling out ations that are stritly dominated given that players are
onstrained to hoose ations in
equilibrium,
i
Ãihin
The interpretation of these objets is that some player
an equilibrium. Contrats onstrain the players to hoose from the sets
Ã1 , . . . , Ãm .
Finally, onsistent with the idea of perfet Bayesian
might ontain only ations that are onsistent with Bayesian equilibrium in the
If mixing is allowed in the last stage, then the support of sij (fi , t−j ) should be ontained in pij (t−i ).
20
MICHAEL PETERS AND BALÁZS SZENTES
game dened by subsets
some
Ã1 , . . . , Ãm
, ommon belief that the non-deviators' types lie in
belief about the deviator i. We will all a Bayesian equilibrium in ontrats an
T̃−i
and
R-equilibrium
if following eah array of ontrat oers for whih player i's ontrat is inonsistent with his equilibrium strategy and all other players ontrat oers are onsistent with the equilibrium strategies
of players whose types lie in
T̃−i ,
ontinuation play lies in
i
h
Ri Ã1 , . . . , Ãm , T̃−i
where the
Ãj
are
the ontratual ommitments of eah of the players. We an now state our main theorem.
Theorem 3.
An alloation rule
s : T → A an be supported as an R-equilibrium in ontrats if and
r,
only if there is a ommitment orrespondene
i
punishments
p i=1,...,m , and
Q
τ i , eah pi is
respet to τ =
in
r (t)
for eah
a olletion of information partitions
outome funtions
i
s i=1,...,m
suh that
r
{τ i }i=1,...,m ,
is measurable with
τ −i , the support of s (t) is ontained
t, s (fi , t−i ) ∈ Ri fi (t−i ) , p (t−i ) , τ −i (t−i ) for eah fi and t−i , and (5.1) and
i
(5.2) are satised.
measurable with respet to
i
One of the key properties of this theorem is to show that when ontrats are denable, equilibrium must support a single punishment orrespondene of exatly the kind we have desribed.
Speially, it is a punishment orrespondene that is independent of
f.
This is entral to the
reiproity idea that we developed at the beginning of the paper. Unooperative behavior by one
player provokes a punishing ontratual response from the others that doesn't depend on exatly
how the deviator goes about being unooperative. We want to show that it is the seond stage
ommitments that apture this property of reiproity.
7
This property of ontrat equilibrium is
also very surprising. It might seem that ontrat equilibrium ould be supported with very ompliated ontrats that punish deviators in a way that is sensitive to exatly how they deviate. We
show that this is not the ase.
Finally, the theorem is written assuming that players use pure strategies at every stage. The
primary reason we assume away mixed strategies is so that we an use an information partition to
represent players knowledge at the seond stage instead of a more omplex measure of information.
We assume pure strategies in the seond stage simply for onsisteny. It is ompletely trivial to
extend the theorem to allow randomization at the seond stage. This involves nothing more than
assuming that
ui (s, t)
sii (fi , t−i ) are mixtures on Aj
whose support is ontained in
to be expeted utility assoiated with the mixture
s
when
pjj (t−i ), then redening
s ∈ △ (A).
The theorem and
proof then proeed verbatim.
6.
We write the proof in three parts.
Proof of Theorem 3
The rst part shows the 'if ' part of the theorem.
generalization of the reiproal ontrating idea presented above.
It is a
Before going on to the more
diult 'only if ' part, we prove the Lemma that is interesting for its own sake, and whih forms
the basis of the seond part of our proof. Finally, we give the proof of the only if part.
7
Care here is needed to observe that seond stage
behavior
does depend on the deviation in the rst stage.
DEFINABLE AND CONTRACTIBLE CONTRACTS
6.1.
21
If Part:
Proof.
Suppose that
s, τ , r, pi , si
satisfy (5.1) and (5.2). We onstrut a Bayesian equilib-
t
x denote xjj
rium in the ontrating game whih implements the alloation s. Let
where
t
xjj
j∈{1,...,m}, tj ∈Tj
denotes a free variable. Consider the following ontrat in
=
|T |
,
free variables:
ctxi ([c1 ] , ..., [cm ])

tk
tk

< xtkk >(x) = [ck ] ,
if ∀k∃xk ∈ xk : tk ∈ Tk
s.t.
 ri (t)

pji (t−j ) if k : ∄xtkk ∈ xtkk : tk ∈ Tk s.t. < xtkk >(x) = [ck ] = j,



Ai
otherwise and if k + 1 > k if k ∈ {j : τ (tj ) = τ (ti )} ,
The last statement is in the third line is always true. Suh a statement, however, makes it possible
that a player with two dierent types oers two dierent but omputationally equivalent ontrats.
Let
γ tii
denote the Godel Code of this ontrat and let
oered by player
i
t
with type ti will be: cγi . Then
=
Notie that
γ = γ tii
i,ti
.
The equilibrium ontrat
ctγi ([c1 ] , ..., [cm ])


if ∀k∃tk ∈ Tk s.t.
< γ tkk >(γ) = [ck ] ,

 ri (t)
pji (t−j ) if k : ∄tk ∈ Tk s.t. < γ tkk >(γ) = [ck ] = j,



otherwise and if k + 1 > k if k ∈ H (ti ) ,
Ai
t
t
< γ qq >(γ) = cγq .
Therefore, the previous ontrat an be rewritten as
ctγi ([c1 ] , ..., [cm ])

t 
if ∀k∃tk ∈ Tk s.t.
cγk = [ck ] ,

 ri (t)
t j
=
pi (t−j ) if k : ∄tk ∈ Tk s.t. cγk = [ck ] = j,



otherwise and if k + 1 > k if k ∈ H (t ) ,
A
(6.1)
i
i
Next, we speify the strategies of the players at the seond stage. If for all
suh that player
j
tj
oers a ontrat cγ , then Player
player deviated, say Player
Dene
fk : T−k → 2
Ak
k,
takes ation
and he oered a ontrat
ck ,
si (t).
and player
j
there is a
tj ∈ T j
Suppose now that one
oered
t
cγj
for all
j 6= k .
as follows:
fk (t−k ) = ck
(6.2)
i
j
h i
t
cγj
ck , ctγj j6=k ,
denotes the vetor of the Godel odes of players other than k . Dene player i's
j6=k
k
strategy as si (fk , t−k ). Notie that by (6.1) these seond-stage strategies are onsistent with the
k
restritions imposed by the ontrats and the renement Rk , that is, si (t) ∈ ri (t) and s (fk , t−k ) ∈
Rk fk (t−k ) , pk (t−k ) , τ −k (t−k ) . (We do not have to speify the strategies if more than one
where
players deviate at the ontrating stage.)
We shall argue that the strategies desribed above onstitute an
R−equilibrium in the ontrat-
m
ing game. First, we show that the strategies {si }i=1 are optimal in the seond stage. Consider
22
MICHAEL PETERS AND BALÁZS SZENTES
onstraint (5.1) with
t′i . This onstraint requires
ti =
of the other players.
to be a best response to the strategies
It remained to show that players do not have inentive to deviate at the
k
ontrating stage. Suppose that player
ctγk .
si (t)
We shall onsider two ases.
tk oers a ontrat ck whih is dierent from
t′k
cγ but τ k (tk ) 6= τ k (t′k ). Then, by (5.1), this
with type
Case 1:
ck =
deviation is not protable no matter what the strategy of player
t′
ck 6= cγk
for all
t′k ∈ Tk .
Suh a deviation indues player
i
k
is at the seond stage. Case 2:
with type
ti
to take ation
ski (fk , t−k ).
Hene, by (5.2) suh a deviation annot be protable.
6.2.
Invariant punishment orrespondene.
of the punishment orrespondene
denes
fi .
p
i
The point of this setion is to show the existene
. A deviator ontemplates dierent ommitment orrespon-
What this Lemma shows is that there has to existene some xed punishment orre-
spondene
pi (t−i )
suh that no matter whih ommitment orrespondene the deviator want to
implement, there must be a way for him to write his ontrat in suh a way that the response of the
non-deviators is exatly the same, and is given by this orrespondene
pi .
This is a onsequene
of the fat that ontrats are required to be denable funtions.
Let
ctii
denote the ontrat of Player
Lemma 4.
For any array
i
funtions, pk
(t−i )
for all
ctii
k 6= i,
i
with type
i=1,...,m, ti ∈T i
ti .
Dene
o
n
t′
τ (t) = t′ ∈ T : ∀i ctii = cii .
of ontrats and every
suh that for any
τ −i
i,
there are
measurable funtion
τ −i
measurable
fi : T−i → 2Ai
, there
∗
is a ontrat ci suh that
h i
t
c∗i [c∗i ] , cjj
(6.3)
and for all
k 6= i
ctkk
(6.4)
h i
t
[c∗i ] , cjj
= f (t−i )
= pik (t−i ) .
j6=i
j6=i
t
cjj .
Eah
alternative ontrat that he oers against this array indues a ommitment orrespondene
f (t−j )
In a ontrat equilibrium, a player expets his opponents to oer the ontrats
and eliits some kind of response. The Lemma shows that provided the ontrats the others oer
are all denable funtions, there must exist some olletion of punishment orrespondenes
suh that for any ommitment orrespondene that player
i
wants, he an write his own ontrat
in suh a way that the others respond with exatly the same punishment
First, we reformulate the statement of the lemma. Let
sional vetor of subsets of
|T |
(Ai )τ −i
For all
n
τ
=
τ (t )
Ai −i −i
A−i−i
t−i
t−i ∈T−i
τ
⊂ (Ai )
: A−i−i
(t−i )
|T−i |
(Ai )τ
:
t
Ai−i
dene
⊂ A−i , ∃ci
S
∈
t
Ai and Ai−i
τ (t )
Ai −i −i
s. t.
=
t−i
t′
Ai−i if
pik
.
denote the set of
whih are measurable with respet to
|T−i |
t−i
(t−i )
t
Ai−i
Ai
i
pk
τ −i ,
|T−i |
dimen-
that is,
τ −i (t−i ) =
τ −i t′−i
.
as follows:
o
i
h
i
h
τ (t )
t−i
t−i
τ (t )
t−i
= A−i−i −i .
[ci ] , c−i
= Ai −i −i , c−i
ci [ci ] , c−i
DEFINABLE AND CONTRACTIBLE CONTRACTS
τ −i (tt−i )
A−i
Ai −i
n
o
τ (t )
S
Ai −i −i
∈S
τ
(t−i )
23
. By the denition of S ,
t−i
t−i
there exists a ontrat, ci , available for player i suh that if the type prole of the other players
τ −i (t−i )
is t−i , then if player i oers ci then his restrited ation spae will be Ai
and players −i's
τ −i (t−i )
. We laim that the statement of the lemma is equivalent
restrited ations spae will be A−i
Let us explain what it means that
to
∩ τ −i (t−i ) ff
Ai
(6.5)
t−i
t−i
6= {∅} .
To see this, suppose rst that the previous displayed statement is true, and
an element of the intersetion.
Dene
pki (t−i )
τ −i (t−i )
to be A−i
for all k
6= i
τ (t )
A−i−i −i
and
pki
t−i
is
t−i ∈ T−i .
(t−i ) t−i ∈
τ −i measurable funtion fi : T−i → Ãi , onsider S (fi (t−i ))t−i . Sine
S (fi (t−i ))t−i , and by the denition of S , there exists a c∗i suh that (6.3) and (6.4) are satised.
k
τ measurable funtions
Conversely, suppose that (6.5) is not true. Then, for all pi (t−i )
k6=i −i
o
n
τ −i (t−i )
|T |
∈ (Ai ) −i , suh that
there exists Ai
For a
t−i
pii (t−i ) = pki (t−i )
n
o
τ −i (t−i )
p−i
(t
)
∈
/
S
A
−i t−i
i
i
t−i
. Then if fi (t−i ) is dened to be
k6=i
∗
does not exist a ontrat ci suh that (6.4) is satised.
where
Proof.
Suppose by ontradition that ∩ τ (t ) ff
Ai −i −i
o
|T−i |
τ (t )
⊂ A−i
A−i−i −i
t−i
n
o τ −i (t−i )
S
Ai
. Let us
n
t−i
o
n
τ (t )
∀ A−i−i −i
Let
cx
fτ −i (t−i )
there exists an
x a funtion
t−i
denote the projetion of
f
τ
Ai −i
(t−i )
t−i
⊂ Ai
o
n
τ (t )
: A−i−i −i
t−i
orresponding to
t−i
|T−i |
i |T−i |
−i |T−i |
→ 2(A )
f : 2(A )
−i |T−i |
⊂ A
,
n
o
τ (t )
S
Ai −i −i
o
n
τ (t )
Ai −i −i
t−i
suh that
t−i
Sine
f
and
cτ −i (t−i )
are denable funtions,
cx
o
n
τ (t )
A−i−i −i
t−i
. Dene
c = cτ −i (t′ ) ,
−i
otherwise.
is a denable funtion in one free variable. Let
denote its Godel ode. Then

 f
cτ −i (t−i ) ([cγ ]) t
τ −i (t′−i )
−i
cγ (c) =

Ai
∈
/
.
−i
st.
there
Then, for all
f = fτ −i (t−i ) t
That is,
∃t′−i ∈ T −i
t−i ∈ T−i ,
= {∅}.
n
o
t−i
∈
/S f
A−i
t−i .
if
suh that
as follows:

 f ′ c
t−i
τ −i (t−i ) ([< x > (x)]) t−i
cx (c) =

Ai
for all
if
∃t′−i ∈ T −i
st.
c = cτ −i (t′ ) ,
−i
.
otherwise.
γ
24
MICHAEL PETERS AND BALÁZS SZENTES
Notie that
cτ −i (t−i ) ([cγ ]) t
(6.6)
−i
S . On the other hand,
=
cγ cτ −i (t−i )
t
by the denition of
=
cτ −i (t−i ) ([cγ ]) t
−i
f.
cγ
cτ −i (t−i )
t−i
−i
and therefore,
by the denition of
n
o cτ −i (t′ ) ([cγ ]) ′
ft−i
−i
t−i
t−i
,
f cτ −i (t−i ) ([cγ ]) t
−i
(6.7)
∈S
∈
/S
cγ
cτ −i (t−i )
t−i
Notie that (6.6) and (6.7) ontradit to eah others, and hene the (6.5)
holds.
6.3.
Only if part of the proof of Theorem 3.
Proof.
m
i m
,
Fix an equilibrium in the ontrating game. We shall onstrut the objets τ , s, {ri }i=1 , s
i=1
i m
and p
suh that the onstraints (5.1) and (5.2) are satised. Denote the equilibrium ontrat
i=1
ti
of Player i with type ti by ci . Dene the partition, τ , as follows:
o
n
t′
τ (t) = t′ ∈ T : ∀i ctii = cii .
m
Next, we onstrut the funtions
{ri }i=1 .
(6.8)
ri (t) = ctii
for all
i ∈ {1, ..., m}.
the denition of
τ.
Notie that
Let
ri (t) ∈ 2Ai .
h t i
ctii , cjj
j6=i
In addition,
ri
,
is measurable with respet to
stage strategy of Player
i
qiti
h i
t
cjj
j6=i
ti . Observe that
h i ti h tj i
tj
ti
ti
∈ ci
ci , cj
cj
qi
with type
(6.9)
j6=i
j6=i
and (6.9). Let
s (t)
is obviously measurable with respet to
denote
τ −i .
denote the seond
must be satised aording to the rules of the ontrating game. Dene
si (t)
by
The seond-stage strategies depend on the ontrats oered at the rst stage.
First, we deal with strategies on the equilibrium path. Let
The funtion
τ −i
si (t) to be qiti
In addition,
h i
t
cjj
j6=i
si (t) ∈ ri (t)
.
by (6.8)
(s1 (t) , ..., sm (t)).
(τ , {r
} , s) satisfy
(5.1). First, onsider this onstraint with
ih
i t
t
cjj
requires qi i
to be a best-response of player i to the
We are ready to show that the triple
t′i = ti .
Then, this onstraint
strategies of the other players. Sine
and hene, (5.1) is indeed satised.
requires player
i
j6=i
qiti
was an equilibrium strategy, it has to be a best response
Seond, onsider (5.1) with
t′i 6= ti .
ti
with type ti to prefer to oer ontrat ci instead of
t′
ci i .
Then, this onstraint
Indeed, the left-hand-side
is just his equilibrium payo and the right-hand-side is the maximum payo of player
i
with type
DEFINABLE AND CONTRACTIBLE CONTRACTS
ti
t′i
ci
if he oered
. Sine,
ctii
was an equilibrium ontrat, suh a deviation annot be protable
and hene, (5.1) is satised.
It remains to onstrut
for all
k 6= i
and for all
m
{pi }i=1
and
i ∈ {1, ..., m}
denote the ontrat of Player
i
k 6= i
ctkk
i
Let qk
h i
t
fi , cjj
j6=i
ally deviates to ontrat
i m
s i=1
and show that (5.2) is also satised. Dene
aording to the statement of Lemma 4. In addition, let
h i h i
t
cfi i , cjj
j6=i
h
i h i
t
cfi i , cjj
j6=i
measurable with respet to
cfi i
= fi (t−i )
= pki (t−i ) .
denote the o-equilibrium strategy of player
cfi i .
pik (t−i )
suh that
cfi i
and for all
25
Dene
sik (fi , t−i )
τ −ik (t−ik ).
to be
h i
t
i
qk fi , cjj
k when
j6=i
.
player
i
unilater-
The funtion
Given these notations, (5.2) requires that player
i
sik
is
an-
fi
not protably deviate by oering an o-equilibrium ontrat in the form of ci , and hene, this
i
onstraint is satised. Finally, sine qk is an o equilibrium strategy in an R-equilibrium, so
h i
t
qki fi , cjj
j6=i
Ri
cfi i
h i h i
t
cfi i , cjj
j6=i
t−i
, c−i
∈
k6=i
h i h i
t
cfi i , cjj
j6=i
, τ −i (t−i ) =
Ri [fi (t−i ) , pi (t−i ) , τ (t−i )]
and the renement ondition is satised.
6.4.
Example 1 - making ations depend on types.
The following examples illustrate the
properties of the ontrating equilibrium in the Bayesian ase. The rst example, illustrates how
ontrating equilibrium an be used to make one player's ation depend on another player's type.
This is something that annot be aomplished in the Bayesian equilibrium of the original game.
In this example, the row player is privately informed and has one of two equally likely types,
and
t2 .
player,
Eah player has two possible ations in the default Bayesian game,
{b1 , b2 }
{a1 , a2 }
t1
for the row
for the olumn player. The payos for eah of the row player's types are given in
the following tables:
b1
b2
b1
a1
3, 3 −1, 4 a1
a2
0,0
0, 0
a2
0,0
b2
0,0 .
−1, 4 3, 3
This is a relatively simple oordination problem, save two things - the way the players want to
oordinate depends on the row player's type, and if the olumn player learns the row player's type,
his weakly dominate ation is inonsistent with the oordinated outome. The unique Bayesian
equilibrium has player 1 using ation
a1
if his type is t1 and ation
a2
if his type is t2 . The olumn
26
MICHAEL PETERS AND BALÁZS SZENTES
player randomizes with equal probability between his two ations. The expeted payos to the
3
olumn player are 2 , the expeted payo to the row player is 1 for eah of his types.
s (t1 ) = (a1 , b1 )
A mehanism designer an implement the oordinated outome
(a2 , b2 )
and
s (t2 ) =
by simply asking the informed agent his type, then instruting the uniformed agent whih
of his ations to take. If either player refuses to partiipate, then they simply play the Bayesian
equilibrium desribed above. The alloation is inentive ompatible and individually rational from
the mehanism designer's perspetive.
To show that the alloation rule
mitment orrespondene
s
is implementable as a ontrat equilibrium, dene the om-
r1 (t1 ) = {a1 }, r1 (t2 ) = {a2 }, r2 (t1 ) = {b1 }, r2 (t2 ) = {b2 }.
This
ommitment orrespondene is measurable with respet to full information and implements the
alloation
s
sine players never have any hoies to make ex post. It is inentive ompatible so
it will be implementable if there is a type ontingent punishment that the row player an impose
8
that makes it unprotable for the olumn player to try to exploit this type information.
evidently the punishment
p1 (t1 ) = {a2 }
and
p1 (t2 ) = {a1 },
This is
sine this holds the olumn player's
payo to zero no matter what he does.
It might help at this point to desribe the way the ontrat equilibrium works in this example.
The informed player writes a dierent 'reiproal' ontrat for eah of his possible types. These
ontrats both speify the same target Godel ode, say
the Godel ode of the uninformed player's ontrat is
ation
a1 .
n
∗
n∗ .
The ontrat for type
t1
says that if
, then the informed player will ommit to
If the Godel ode of the uninformed player's ontrat is anything else, then the informed
player of type
t1
will ommit to ation
a2 .
The ontrat for t2 is similar with the ations reversed.
Enoding these ontrats gives a pair of Godel odes, say
m1
and
m2 ,
orresponding to eah of
the informed player's possible ontrats. The uninformed player writes a ontrat that says that
if the Godel ode of the informed player's ontrat is
ode of the informed player's ontrat is
to
{b1 , b2 }
integers
6.5.
m2 ,
m1 ,
then he will ommit to
then he will ommit to
b2 ,
b1 ,
if the Godel
otherwise he will ommit
and hoose among them ex post. The theorem above shows that there is a triple of
∗
(n , r1 , r2 )
suh that the Godel ode of the uninformed player's ontrat is
n∗ .
Example 2: ontrat equilibrium doesn't do as well as a mehanism designer.
In the example just desribed, the ontrat equilibrium supports everything that a mehanism
designer might want to implement. However, as we mentioned in the introdution to this setion,
ontrat equilibrium imposes a restrition on feasible alloations. When a player deides to deviate,
he knows that he will learn something about the types of the other players when he sees their
ontrats. In addition, sine he an ondition his behavior on ontrats, he an make his deviation
depend on this type information.
To illustrate the limitations that this imposes, onsider the following variant of the example
given above.
8
There are again two players eah with two possible ations.
The row player has
Of ourse, the uninformed player must also speify a punishment. For simpliity, we speify it below.
DEFINABLE AND CONTRACTIBLE CONTRACTS
two possible types, either
t1
or
t2 ,
whih are equally likely.
27
The olumn player has no private
information. The payos for eah of the informed player's possible types are given in the following
tables:
b1
b2
b1
a1
3, 3 −1, 4
a2
0, 4 2, −1
and
b2
a1
2, −1 0, 4
a2
−1, 4 3, 3
.
The Bayesian equilibrium of this default game has eah player randomizing with equal probability over eah of his ations no matter what his information. The informed (row) player has payo
5
1 in this equilibrium no matter what his type, while the uninformed olumn player has payo 2 .
The Myerson mehanism designer has no problem implementing the alloation
s (t2 ) = (a2 , b2 ).
s (t1 ) = (a1 , b1 ) and
He does this by inviting the players to partiipate in a mehanism in whih he
asks the row player to report his type. If he reports t1 then he instruts the players to use ations
a1
and
b1 ,
and similarly when type
t2
is reported. By agreeing to partiipate, the players ommit
themselves to follow the mehanism designer's instrution. This is inentive ompatible beause
the row player's payo falls from
3
to
2
if he misreports his type. The alloation is individually
rational in the usual mehanism design sense as long as a refusal to partiipate by either player
results in both players playing the (unique) Bayesian equilibrium of the original game.
This alloation rule annot be implemented as a ontrat equilibrium. Aording to Theorem 3,
to implement it, there must be a type ontingent ommitment that the row player an make, and
some speiation of the row player's ations ex post that hold the olumn players payo below
3
when he simply ommits to hoose his from his possible ations
there is no suh punishment, observe that if
ation
a1
π
b1
and
b2
ex post. To see that
is the probability with whih the row player uses
in the ex post game, then the olumn player's payo when the row player has type 1 is
max [π3 + (1 − π) 4, π4 − (1 − π)] =
max [4 − π, 5π − 1] ≥
19
.
6
The argument is idential when the row player has type 2. No suh punishment exists. Thus by
Theorem 3, there is no ontrat equilibrium that supports this outome.
As before, if there were a ontrat equilibrium that ould support this outome, then the olumn
player has to take an ation that depends on the row player's type. In priniple, he an do this
beause he an ommit himself to an ation that depends on the row player's ontrat, whih in
turn depends on the row player's type. However if the olumn player knows that the ontrat will
reveal the row player's type, then a deviation to a ontrat that simply allows the olumn player
to take his ation ex post has to be protable.
6.6. Payos lie between the Bayesian equilibrium and those implementable by a mehanism designer. This nal example is intended to illustrate a number of things. First, it shows
that the ontrat equilibrium implements stritly more than the Bayesian equilibrium of the default
28
MICHAEL PETERS AND BALÁZS SZENTES
game, but less than what is implementable by a mehanism designer. Seond, it has non-degenerate
ommitment and punishment orrespondenes, both of whih are type dependent.
The example also illustrates how randomization an be inorporated into the nal stage of the
ontrating proess when players hoose ations from their ommitment orrespondenes. We have
ignored randomization in the statement of our main theorem to simplify. This example illustrates
how the extension works.
The row player has three equally likely types supporting payos given in the following tables:
t1
b1
b2
t2
b1
b2
t3
a1
3, 3 −1, 4
a1
2, −1 0, 4
a1
a2
0, 4 2, −1
a2
−1, 4 3, 3
a2
b1
b2
−2, −2 4, −1
0, 4
2, 11
4
The payos in the rst two boxes are the same as they were in the seond example disussed
above.
The unique Bayesian equilibrium for this game has the uninformed player randomizing
equally between
b1
and
b2 .
The informed player randomizes equally when his types are
t2 , but hooses a1 with probability 59 when his type is
t3 .
and
The payo to the informed player is
no matter what his type, while the payo to the uninformed player is
designer an implement the alloation rule
t1
2.
1
A Myerson mehanism
s (t1 ) = (a1 , b1 ), s (t2 ) = s (t3 ) = (a2 , b2 )
exatly as he
does in the rst example, by asking the informed player his type, then telling both players what
ations to take. This alloation an't be supported as a ontrat equilibrium. The argument is
exatly as in the seond example above, sine the ontrat equilibrium has to enfore dierent
ations when the row players types are
t1
and
t2 .
However, the alloation in whih both players randomize equally between their ations when
the row player has type 1 or type 2, while the ations
a2
b2
and
are taken when the row player
has type 3 an be supported as a ontrat equilibrium. This alloation is measurable with respet
to the information partition
{{t1 , t2 } , {t3 }}
so the ommitment and punishment orrespondenes
an depend on the row player's type. In partiular, the ommitment orrespondene we want is
r1 (t1 ) = r1 (t2 ) = {a1 , b1 }, r1 (t3 ) = {a2 },
while
r2 (t1 ) = r2 (t2 ) = {b1 , b2 } and r2 (t3 ) = {b2 }.
punishment orrespondene for the row player is again multi-valued
while
p1 (t3 ) = {a1 }.
The olumn player punishes with
{b1 , b2 }
The
p1 (t1 ) = p1 (t2 ) = {a1 , a2 },
for all deviations.
The behavior to be supported involves randomization among the hoies in the ommitment
orrespondene. This an be inorporated in a straightforward way by requiring that the mappings
s (t)
si (t−i )
and
have their range in the set of mixtures over ations whose support lies within the
appropriate ommitment orrespondene.
si (t3 ) = {0, 1}
for
i = c, r.
So in the example,
eah
f,
and
s (f, t3 ) = {1, 0}
{{t1 , t2 } , {t3 }}.
1
2, 2
= si (t2 )
Let
sc (f, t1 ) = sc (f, t2 ) =
as is required by the punishment orrespondene.
is a type ontingent ommitment
partition
1
while
The behavior during the punishment phase is dened similarly.
Consider the ase where the olumn player deviates.
c
si (t1 ) =
f
1
1
2, 2
for
A deviation
that has to be measurable with respet to the information
As an example, take
f (t1 ) = {b1 , b2 } = f (t3 )
while
f (t3 ) = {b1 }.
The
DEFINABLE AND CONTRACTIBLE CONTRACTS
29
punishment has to make this and all other measurable type ontingent ommitments unprotable
given the olumn player's interim beliefs. It is straightforward to hek that it aomplishes this.
One way to implement this in a ontrat equilibrium is to have the row players types
both oer the same ontrat whih ommits them to
oers. When the row player has type
t3
{a1 , a2 } whatever ontrat the
he oers a ontrat that ommits him to
n∗c ,
a2
t1
and
t2
olumn player
if the olumn
a1 against
∗
any other Godel ode. The Godel ode of this ontrat is, say n3 . The olumn player oers a
∗
ontrat that ommits to b2 if the Godel ode of the row player's ontrat is n3 , but ommits to
player oers a ontrat whose Godel ode is equal to some target
{b1 , b2 }
but ommits to
against any other ontrat.
7.
Independent Private Values
An environment that is of some interest in appliations is the independent private value environment. The lassi rst or seond prie aution models are typial examples. However, the
tratability of suh models makes them popular. For the independent private value environment
we an use our theorem to provide something that looks like a folk theorem for Bayesian equilibrium. For our purposes, this 'folk theorem' is interesting beause it suggests an environment where
ontrats an be used to fully deentralize the mehanism designer's problem.
Players have
private values
if
ui (a, (ti , t−i )) = ui a, ti , t′−i
(players' payos don't depend on other players' types).
E (f (t−i ) : ti ) = E (f (t−i ) :
for all
a ∈ A, t−i , t′−i ∈ T−i
Types are independently distributed if
t′i ) for every i, ti , t′i and every integrable funtion
f.
Theorem 2. Let s : T → A be an alloation rule that is implementable by a entralized mehanism
designer in an independent private value environment. Then the alloation s an be supported as
a Bayesian equilibrium in ontratable ontrats.
Proof.
An alloation rule is implementable by a mehanism designer if there is a olletion of
punishments
si : T−i → A−i ,
where
sij
is the punishment partiipants will impose on player
i
if he
′
hooses not to partiipate suh that for eah ti and ti
Et−i
and
Et−i (ui (s (t) , t) : ti ) ≥
ui ai , s−i t′i , t′−i , ti , t′−i : ti ;
Et−i (ui (s (t) , t) : ti ) ≥
max Et−i ui ai , si (t−i ) , (ti , t−i ) : ti .
ai ∈Ai
We prove the theorem by onstruting the various omponents required by Theorem 3.
Begin with the punishment
si (·).
We have from the private value and independene assumption
max Et−i ui
ai ∈Ai
ai , si (t−i ) , (ti , t−i ) : ti =
max Et−i ui
ai ∈Ai
ai , si (t−i ) , ti .
30
MICHAEL PETERS AND BALÁZS SZENTES
Let
g̃
i
be the distribution on
A−i
indued by the funtion
si
and the distribution of t−i and dene
the punishment
s̃i (f, t−i ) = g̃ i
for eah
t−i
and every orrespondene
f : T−i → Ãi
that is measurable with respet to full
information (a set whih ontains all orrespondenes whih are measurable with respet to any
information struture). Then we have
Et−i (ui (s (t) , t) : ti ) ≥
max Et−i ui ai , s̃i (fai , t−i ) , ti =
ai ∈Ai
max E ui
ai ∈Ai
where
fai (t−i ) = {ai } ∀t−i ∈ T−i .
depend on
t−i .
ai , g̃ i , ti
The important aspet of this punishment is that it does not
f
Let τ i be the full information partition of
Ti
with
τf =
Q
i
τ fi .
Dene the
i
ommitment orrespondene ri (t) = {si (t)} and the punishment orrespondene pj (t) = Aj .
f
These orrespondenes are both trivially measurable with respet to τ . Furthermore, (5.1) holds
trivially sine the alloation must be inentive ompatible, and
any ommitment orrespondene
Et−i
f
ri
is always a singleton. Now for
measurable with respet to the full information partition
max Et′−i ui a, s̃
a∈fi (t−i )
i
fi , t′−i
max E ui
a∈Ai
, ti :
t′−i
a, g̃ i , ti
∈ τ −i (t−i ) : ti
τ f−i ,
≤
≤
Et−i (ui (s (t) , t) : ti ) .
So (5.2) is satised. Then by Theorem 3, the alloation
s is supportable as a ontrat equilibrium.
From the proof above, it should be apparent that what makes the theorem work is the fat that
the punishment that the mehanism designer uses to enfore partiipation has the same impat on
the non-partiipant no matter what he learns about the partiipants' types. This is a onsequene
of the private value assumption. Some interdependent value problems will also have this property.
For example, in a trading problem, not being able to trade may be worse for a player than trading
no matter what he learns about the types of the others. Similar folk theorems are possible in suh
environments.
8.
Conlusion
This paper shows how the ontrats on ontrats approah an be extended to environments
with inomplete information by restriting players to use denable ontrats. Denable ontrats
onstitute the largest lass of arithmeti ontrats whih an be written as a nite text in a rst
order language. In this sense denable ontrats embed most other interesting lasses of feasible
ontrats as subsets.
DEFINABLE AND CONTRACTIBLE CONTRACTS
31
In ontrast to the omplete information ase, we show that the 'folk theorem' doesn't generally
hold.
A entralized mehanism designer an implement alloations that an't be supported as
equilibrium with ontratible ontrats. This limitation is not a onsequene of the set of feasible
ontrats, but rather of the fat that publi ontrats reveal information about non-deviators'
type.
The restrition to denable ontrats allows us to provide a omplete haraterization of
equilibrium and to prove this result. One of the results we provide as part of our main theorem
illustrates the role that punishments play in a stati ontrating environment.
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