ECON 3210/4210 Decisions, Markets and Incentives
Lecture notes 2.11.05
Nils-Henrik von der Fehr
STATIC GAMES OF INCOMPLETE
INFORMATION
Introduction
Complete information: payoff functions are common knowledge
Incomplete information: at least one player is uncertain about another player’s
payoff function
„
Bayesian games
Examples
„
(sealed-bid) auctions
„
bargaining (dynamic)
Representation
„
normal form or extensive form
„
beliefs
Solutions: Refinement of Nash Equilibrium
„
Bayesian Nash Equilibrium
„
beliefs at equilibrium
Normal-Form Representation
The normal-form representation of an n-player static Bayesian game specifies
the players’ strategy (action) spaces S1,...,Sn ( A1,..., An ), their type spaces
T1,...,Tn , their beliefs p1,..., pn and their payoff functions u1,..., un . The game is
denoted G = {S1,..., Sn ;T1,...,Tn ; p1,..., pn ;u1,...,un } .
Player i’s type, t i , is privately known by player i, determines player i’s payoff
function ui ( s1,..., sn ; t i ) and is a member of the set of possible types, Ti .
Example
„
high-cost versus low-cost firms
„
successful versus unsuccessful developers of new technology
Player i’s belief pi ( t − i t i ) describes i’s uncertainty about the n-1 other players’
possible types, t − i , given i’s own type, ti . The function pi ( t − i t i ) gives the
probability (as seen by player i) that opponents’ types are t − i , conditional on
player i’s type being ti .
Example
„
probability distribution over competitors’ costs
„
probability distribution over whether competitors have succeeded
Note: in both cases beliefs may depend on own type (eg. more likely that
competitor has low costs (has been successful), if the firm itself has low costs
(has been successful)).
Problem: How are beliefs determined? How do we ensure that beliefs are
compatible?
Timing (Harsanyi, 1967)
1)
Nature draws a type vector t = ( t1,..., tn ) , where ti is drawn from the set
Ti .
2)
Nature reveals t i to player i, but not to any other player.
3)
Players simultaneously choose strategies (actions).
4)
Payoffs are received.
Note that this set up transforms a game of incomplete information into a game
of imperfect information – the move of Nature is not (perfectly) observed and
hence players do not know the whole history of the game when they have to
make their move.
It is assumed that the probability distribution according to which Nature moves
is common knowledge; that is, all players know that Nature draws
t = ( t1,..., t n ) according to the probability distribution p ( t ) .
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After individual types have been revealed, individual beliefs can be calculated
using Bayes’ rule:
pi ( t − i t i ) =
p ( t − i , ti )
p ( ti )
=
p ( t − i , ti )
∑ p (t )
i
t − i ∈T− i
where p ( t − i , t i ) is the probability that nature draws type t i for player i and
types t − i for i’s opponents, and p ( t i ) is the (marginal) probability that nature
draws type ti for player i.
It follows that other players can compute the various beliefs that player i may
hold, depending upon their types.
Example: Two players, 1 and 2, play a game. Player 2 may be of one of two
types, t = −1 or t = 1 . The probability that Player 2 is of type t = −1 is p. Given
Player 2’s type, the game may be described by the following table:
Player 2
Player 1
L
R
U
1,t
-2,0
M
0,2
0,2
D
-2,0
1,-t
The normal-form representation may be described as follows: Players’
strategy spaces are S1 = {U, M, D} and S2 = {L, R} . Player 1’s type set is a
singleton (the type is common knowledge), while Player 2’s type-space is
T2 = {−1,1} . The beliefs of Player 1 about Player 2’s type is given by
Pr ( t = −1) = p and Pr ( t = −1) = 1 − p . The payoffs are as described in the
table above.
Extension: games in which players have private information not only about
their own payoffs, but about other players’ payoffs also (example: asymmetric
information about demand conditions in oligopoly). Consequently, payoffs are
functions of other player’s types also, u ( s1,...sn ; t1,..., t n ) .
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Bayesian Nash Equilibrium
In a static Bayesian game G = {S1,..., Sn ;T1,...,Tn ; p1,..., pn ;u1,...,un } , a strategy
for player i is a function si ( t i ) , where for each type t i in Ti , si ( t i ) specifies
the action from the feasible set Si that type t i would choose if drawn by
nature.
In a separating strategy, each type chooses a different strategy, i.e.
si tˆi ≠ si ti if tˆi ≠ tj . In a pooling strategy, all types choose the same
( )
( )
strategy, i.e. si ( t i ) = si , all t i ∈ Ti .
In the static Bayesian game G = {S1,..., Sn ;T1,...,Tn ; p1,..., pn ;u1,...,un } the
strategies s * = ( s1 *,...sn * ) are a (pure-strategy) Bayesian Nash equilibrium if
for each player i and for each of i’s types t i , si * ( t i ) solves
max si
∑ p (t
i
−i
t i ) ui ( s1 * ( t1 ) ,..., si ,..., sn * ( tn ) ; t i )
t− i
In other words, given a player’s belief about the other players’ types, his or her
strategy is a best response to the other players’ strategies. This is true
whatever type player i may be.
Example continued: In the above example, Player 2 has a (weakly) dominant
strategy: if his type is t = −1 the best response is R whatever the choice of
Player 1; if his type is t = 1 the best response is L. Suppose Player 1 believes
that Player 2 will play L when he is of type t = −1 – which occurs with
probability p – and play R when he is of type t = 1 – which occurs with
probability 1-p. Then Player 1’s payoff from playing his various strategies are
U : [1 − p ] ⋅ 1 + p ⋅ [ −2] = 1 − 3 p
M : [1 − p ] ⋅ 0 + p ⋅ 0 = 0
D : [1 − p ] ⋅ [ −2] + p ⋅ 1 = 3 p − 2
It follows that U is a best response if p ≤ 31 , D is a best response if p ≥ 32 ,
while M is a best response if 31 ≤ p ≤ 32 . Consequently, we have the following
equilibria
4
1
: s1 * = U, s2 * ( −1) = R, s2 * (1) = L
3
1
2
< p < : s1 * = M, s2 * ( −1) = R, s2 * (1) = L
3
3
2
p > : s1 * = D, s2 * ( −1) = R, s2 * (1) = L
3
p<
When p = 31 and p = 32 we have multiple (i.e. two) equilibria. (Note that, in this
game, Player 2 may actually benefit from the fact that Player 1 does not know
his type; hence he has an incentive to hide his true type, i.e. to make p close
to 0.5.)
First-Price, Sealed-Bid Auction
Suppose two bidders participate in an auction for one object.
Bidder i has valuation (i.e. maximum willingness to pay) v i for the object, i =
1,2. Bidders’ valuations are independently and uniformly distributed on [0,1] .
Bidders are risk neutral.
Bidders simultaneously submit their bids, which must be nonnegative. The bid
of player i is denoted bi , i = 1,2. The bidder with the highest bid wins the
object and pays a price equal to his bid (if bidders submit the same bid, the
object is allocated on a 50-50 basis). The loser gets and pays nothing.
In this game, player i’s strategy space is Si = [0, ∞ ) , while the type space is
Ti = [0,1] . Because valuations are independently drawn, player i’s belief is that
v j is uniformly distributed on [0,1] , no matter what the value of v i is. Player i’s
payoff is
⎧v i − bi
if bi > b j
⎪1
ui ( b1, b2 ;v1,v 2 ) = ⎨ 2 [v i − bi ] if bi = b j
⎪0
if bi < b j
⎩
Suppose bidder i believes bidder j’s strategy – the function from type to
action – is given by b j (v j ) . Let Fj ( b ) denote the probability that bidder j bids
below b. Then bidder i chooses his bid so as to solve the problem
max bi [v i − bi ] Fj ( bi ) .
The first-order condition for this problem may be written
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−Fj ( bi ) + [v i − bi ] Fj′ ( bi ) = 0 .
Note the probability that bidder j bids below b is equal to the probability that
bidder j’s valuation is below v, where v is such that b j (v ) = b . Since
valuations are uniformly distributed on [0,1] , it follows that Fj ( b j (v ) ) = v . By
implicit derivation, we find Fj′ ( b j (v ) ) b′j (v ) = 1 or Fj′ ( b j (v ) ) = 1 b′j (v ) .
We are looking for a symmetric equilibrium, in which both players follow the
same strategy; that is, b1 (v ) = b2 (v ) = b (v ) and F1 ( b ) = F2 ( b ) = F ( b ) .
Substituting into the above equations and solving, we find
−v + ⎡⎣v − b (v ) ⎤⎦
1
=0,
b (v )
which may alternatively be written
b′ ( v ) v + b ( v ) = v .
Note that the left-hand side is equal to
d
dv
( b (v ) v ) . Then, integrating both
sides with respect to v, we find
b (v ) v =
1 2
v +A
2
Note that the bidder type with the lowest valuation must bid zero; that is,
b ( 0 ) = 0 . It follows that
b (v ) =
1
v
2
So, at equilibrium each bidder submits a bid equal to half his valuation.
With n bidders, one can show that strategies are given by
b (v ) =
n −1
v,
n
which implies that bidders bid more aggressively (i.e. closer to their
valuations) the more bidders there are.
The Revelation Principle
See Gibbons, ch. 3.3.
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