Michael Brand
Game Theory
=
The mathematics of
joint decision-making
Game Theory
=
The mathematics of
joint decision-making
Prisoner’s Dilemma
Stag Hunt
(1,1)
(0,5)
(5,0)
(3,3)
Player 1
Player
2
Player
2
Player 1
Battle of the Sexes
(2,3)
(0,0)
(1,1)
(1,0)
(2,2)
Player 1
Player
2
Player
2
*
(3,2)
(0,1)
Chicken
Player 1
(0,0)
(1,1)
(-9,-9)
(-1,1)
(1,-1)
(0,0)
von Neumann and Morgenstern (1944)
(2,3)
ideal(S)
(3,2)
d
(0,0)
S
The utility
feasibility set is
• Convex
• Comprehensive
• Nontrivial
• Closed
• Bounded from
above
The disagreement
point
*
Nash (1950)
S
ψ(S,d) ∈ S ∪ {d}
• feasible
d
*
Nash (1950)




Weak Pareto Optimality (WPO)
Scale Invariance (INV)
Symmetry (SYM)
Invariance to Irrelevant Alternatives (IIA)
arg max
x S , x  d
NBS


  ( xi  d i ) 
 i

IIA
SYM
INV
WPO
d
d
d
*
Nash (1950) + Nash (1953)




Weak Pareto Optimality (WPO)
Scale Invariance (INV)
Symmetry (SYM)
Monotonicity (MONO)
ideal(S)
d
*
Kalai, Smorodinsky (1975)
S
ψ(S,d) ∈ S ∪ {d}
• feasible
d
*
Nash (1950)
S
ψ(S,d) ⊆ S ∪ {d}
•
•
•
•
d
feasible
nonempty
closed
nontrivial
S
ψ(S)
⊆ S⊆ S ∪ {d}
ψ(S,d)
•
•
•
•
feasible
nonempty
closed
nontrivial
d
Harsanyi (1955), Myerson
(1981), Thompson (1981)
*







Individual Rationality
(IR)
Pareto Optimality (PO)
Linearity (LIN)
Upper Linearity (ULIN)
Concavity (CONC)
Individual Monotonicity
(IMONO)
etc...
Axioms






The egalitarian
solution
The dictatorial solution
The serial-dictatorial
solution
The Yu solutions
The Maschler-Perles
solution
etc...
Solutions

The alternating offers game
NBS
d
*
Rubinstein (1982)

Invariance to Irrelevant Alternatives (IIA)
◦ How can we know this, until we consider how the
feasibility set is explored? (Kalai+Smorodinsky)

Weak Pareto optimality (WPO)
◦ The same question applies.
◦ Commodity space may have a different topology.

Symmetry (SYM)
◦ Why, exactly, are we assuming this?
◦ Is life fair?
◦ Are all negotiations symmetric?
Axiom
Quality
Invariance to
Irrelevant Alternatives Thorough
(IIA)
Weak Pareto
Optimality (WPO) /
Benevolent
Pareto Optimality (PO)
Symmetry (SYM)
Impartial




INV claims that by rescaling to other vN-M
utility units, the solution cannot be altered.
It is considered to be a statement regarding
the inability to compare utility
interpersonally.
In fact, it is a stronger statement than this. It
is a claim that all arbitrators must necessarily
reach the same conclusion, because their
decisions must refrain from subjective
interpersonal assessment of utilities.
It is a claim that justice is objective.
To A or to B?
INV
d
Strawberry Shortcake
vs.
Lemon Tart
d
INV
d
d

The most we can require of an arbitrator is
that her method of interpersonal utility
comparisons is consistent.
◦ Or else, again, we are back at the “Strawberry
Shortcake vs. Lemon Tart” dilemma.

SYM now has to be reformulated.
◦ The arbitrator should now be required to be
impartial within her subject world view.

We assume the problem to be scaled into this
world view.


What is the role of d in arbitration?
Is the arbitration binding?
◦ If so: no role.
◦ If not: shouldn’t S reflect real outcomes, as
opposed to apparent outcomes?

This method of modeling actually gives more
modeling power.
(0,4,4)
(4,4,0)
(3,3,3)
(4,0,4)
Note: S no longer
comprehensive.

What division of the cake should John and
Jane decide on, if they are on their way to the
shop and still don’t know which cake is in
store?
CONC
LIN

ex-post efficiency vs. ex-ante efficiency
Axiom
Invariance to
Irrelevant Alternatives
(IIA)
Weak Pareto
Optimality (WPO) /
Pareto Optimality (PO)
Symmetry (SYM)
Concavity (CONC)
Quality
Thorough
Benevolent
Impartial
Uses foresight

WPO+SYM+IIA+CONC ⇔
◦ The Egalitarian Solution or The Utilitarian QuasiSolution (for a comprehensive problem domain)
Utilitaria n ( S )  arg max
x S
x
Egalitaria n ( S )  arg max
x S ,  i , j : x i  x j
Edgeworth (1881),
Walras (1954)
i
i
Zeuthen (1930),
Harsanyi (1955)
x
i
i
Bentham (1907),
Rawls (1971)
Kalai (1977)
“The Veil of
Ignorance”

WPO+SYM+IIA+CONC ⇔
◦ The Egalitarian Solution or The Utilitarian QuasiSolution (for a comprehensive problem domain)

But only the Utilitarian Quasi-Solution ⇔
◦ Admitting non-comprehensive problems
◦ Strengthening WPO to PO
◦ Strengthening CONC to ULIN or to LIN
One of the tenets of the
modern legal system

NBS is a solution, but only when S is
guaranteed to be convex.
◦ Otherwise, it is a quasi-solution, and is known as
the “Nash Set”


The utilitarian quasi-solution is a quasisolution on general convex S. However, it is a
solution on strictly convex S.
A strictly convex S occurs when goods are
infinitely divisible and
◦ Players are risk avoiders; or
◦ Returns diminish




IIA implies that there is a social utility
function
PO implies that this function is monotone
increasing in each axis
LIN implies that it is convex
SYM implies that it maps all coordinate
permutations to the same value
◦ which, together with convexity, leads to being a
function on the sum of the coordinates.

Shapley (1969)’s “Guiding Principle”:
◦ ψ(S) = Efficient(S) ∩ Equitable(S)
IIA
SYM
ULIN
PO(S) ⊆ Efficient(S) ⊆ WPO(S)
The Utilitarian Quasi-Solution
*
On non-comprehensive domains


Now, we do need to look at the mechanics of
haggling.
The mechanics of Rubinstein’s alternating
offers game:
◦
◦
◦
◦
Infinite turns (or else the solution is dictatorial)
Infinite regression of refusals leads to d.
Time costs: <Si+1,di+1>=<(1-ε)Si+ε di,di>
When ε→0, the first offer is NBS and it is
immediately accepted.





<Si+1,di+1>=<(1-ε)Si+ε di,di>
Rubinstein: At each offer, there is a 1-ε
probability for negotiations to break down.
Why should negotiations ever break down for
rational players?
Why at a constant rate?
Is it realistic to assume that no amount of
refusals can ever reduce utility to less than a
fixed amount?
Man, I could
be at home
watching TV
right now...
900
800
400
200
300
700
100
600
I’d rather
be
sailing.




Let A be the vector designating for each
player the rate at which her utility is reduced
in terms of alternate time costs.
St+Δt={x-AΔt|x ∈ St}
We take Δt→0 and tmax→∞.
The result is in the utilitarian quasi-solution
arg max
x S

x
i
i
/ Ai
Note: The mechanics of the bargaining
process dictate the solution’s scaling, with no
need for interpersonal utility comparisons.






W.l.o.g., let us scale the problem to A=1.
We know we are on ∂S (the Pareto surface of S),
and because Δt→0 we know S changes slowly.
Let p(x)=the normal to ∂S at x (the natural rate of
utility exchange).
When backtracking over n turns, the leading offer
changes in direction <1/p1(x)-n/s,... , 1/pn(x)-n/s>,
where s=∑pi(x).
Applying the Cauchy-Schwarz inequality, we get
that ∑xi always increases, except when p(x)∝1.
Letting tmax→∞, we are guaranteed to reach a
point on the utilitarian quasi-solution.



We know this from experience.
We now know that it is rational behavior.
It is not accounted for by NBS (Or
Rubinstein’s alternating offers game).

questions?