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Some simple (not-atypical) games
Prisoner’s Dilemma Stag Hunt
Player 1 Player 1
Player 2
(1,1) (0,5) Player 2
(1,1) (0,1)
(5,0) (3,3) (1,0) (2,2)
Battle of the Sexes Chicken
Player 1 Player 1
Player 2
(0,0) (3,2) Player 2
(-9,-9) (-1,1)
(2,3) (0,0) (1,-1) (0,0)
* von Neumann and Morgenstern (1944)
(1,1)
The bargaining problem
(2,3)
(3,2)
(0,0)
SThe utility feasibility set
d
• Convex• Comprehensive• Nontrivial• Closed• Bounded from above
is
The disagreement point
* Nash (1950)
ideal(S)
* Nash (1950) + Nash (1953)
Weak Pareto Optimality (WPO) Scale Invariance (INV) Symmetry (SYM) Invariance to Irrelevant Alternatives (IIA)
The Nash Bargaining Solution
* Nash (1950)
d
iii
dxSx
dx )(maxarg,
d d
INV
WPO
SYM
IIANBS
* Kalai, Smorodinsky (1975)
Weak Pareto Optimality (WPO) Scale Invariance (INV) Symmetry (SYM) Monotonicity (MONO)
The Kalai-Smorodinsky Solution
d
ideal(S)
Social choice
S ψ(S) ⊆ S
• feasible• nonempty• closed• nontrivial
* Harsanyi (1955), Myerson (1981), Thompson (1981)
d
ψ(S,d) ⊆ S ∪ {d}
An explosion of papers
Axioms Solutions
Individual Rationality (IR)
Pareto Optimality (PO) Linearity (LIN) Upper Linearity (ULIN) Concavity (CONC) Individual
Monotonicity (IMONO) etc...
The egalitarian solution
The dictatorial solution
The serial-dictatorial solution
The Yu solutions The Maschler-Perles
solution etc...
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?
Breaking down the old axioms
Life may not be fair, but a good arbitrator should be
Axiom QualityInvariance to Irrelevant Alternatives (IIA)
Thorough
Weak Pareto Optimality (WPO) / Pareto Optimality (PO)
Benevolent
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.
What about INV?
Does this agree with our intuitive notion of fairness?
d d
INVTo A or to B?
d d
INVStrawberry Shortcake
vs.Lemon Tart
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.
So, good arbitration cannot be “objective”
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.
Bargaining or social choice?
(4,4,0)
(4,0,4)
(0,4,4) (3,3,3
)
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?
We need a new axiom
LIN
CONC
Which brings us to the last quality of a good arbitrator
Axiom QualityInvariance to Irrelevant Alternatives (IIA)
Thorough
Weak Pareto Optimality (WPO) / Pareto Optimality (PO)
Benevolent
Symmetry (SYM) ImpartialConcavity (CONC) Uses foresight
WPO+SYM+IIA+CONC ⇔◦ The Egalitarian Solution or The Utilitarian Quasi-
Solution (for a comprehensive problem domain)
Some of the main results
i
iSx
xSnUtilitaria maxarg)(
i
ixxjiSx
xSnEgalitariaji:,,
maxarg)(
Edgeworth (1881), Walras (1954)
Zeuthen (1930), Harsanyi (1955)
Bentham (1907), Rawls (1971)
Kalai (1977)
“The Veil of Ignorance”
WPO+SYM+IIA+CONC ⇔◦ The Egalitarian Solution or The Utilitarian Quasi-
Solution (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
Some of the main results
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 quasi-
solution 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
Solution or quasi-solution?
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.
A simplified proof for a simplified case
Shapley (1969)’s “Guiding Principle”:◦ ψ(S) = Efficient(S) ∩ Equitable(S)
Beyond the standard axiomatic model
PO(S) ⊆ Efficient(S) ⊆ WPO(S)
SYM ULINIIA
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.
What happens in non-arbitrated scenarios?
<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?
Is this realistic?
Real-life time costs are exogenous to the bargaining problems
900100800200700300600400Man, I could be at home watching TV right now...
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
Note: The mechanics of the bargaining process dictate the solution’s scaling, with no need for interpersonal utility comparisons.
Real-life time costs are exogenous to the bargaining problems
i
iiSx
Ax /maxarg
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.
A simplified proof for a simplified case
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).
Conclusion: Never haggle when you are in a hurry.