Michael Brand
Game Theory = The mathematics of joint decision-making
Game Theory = The mathematics of joint decision-making
Prisoner’s DilemmaStag Hunt Player 1 Player 2 (1,1)(1,1)(0,5)(0,5) (1,1)(1,1)(0,1)(0,1) (5,0)(5,0)(3,3)(3,3)(1,0)(1,0)(2,2)(2,2) Battle of the SexesChicken Player 1 Player 2 (0,0)(0,0)(3,2)(3,2) (-9,-9)(-1,1) (2,3)(2,3)(0,0)(0,0)(1,-1)(0,0)(0,0) * von Neumann and Morgenstern (1944) (1,1)(1,1)
(2,3) (3,2) (0,0) S The utility feasibility set d Convex Comprehensive Nontrivial Closed Bounded from above is The disagreement point * Nash (1950) ideal(S)
S d * Nash (1950) ψ(S,d) ∈ S ∪ {d} feasible
* Nash (1950) + Nash (1953) Weak Pareto Optimality (WPO) Scale Invariance (INV) Symmetry (SYM) Invariance to Irrelevant Alternatives (IIA) * Nash (1950) d dd INV WPO SYM IIA NBS
* Kalai, Smorodinsky (1975) Weak Pareto Optimality (WPO) Scale Invariance (INV) Symmetry (SYM) Monotonicity (MONO) d ideal(S)
S d * Nash (1950) ψ(S,d) ∈ S ∪ {d} feasible
S d ψ(S,d) ⊆ S ∪ {d} feasible nonempty closed nontrivial
S ψ(S) ⊆ S feasible nonempty closed nontrivial * Harsanyi (1955), Myerson (1981), Thompson (1981) d ψ(S,d) ⊆ S ∪ {d}
AxiomsSolutions 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...
The alternating offers game d * Rubinstein (1982) NBS
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?
AxiomQuality Invariance 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.
dd INV To A or to B? dd INV Strawberry 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.
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. (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? LIN CONC
ex-post efficiency vs. ex-ante efficiency
AxiomQuality Invariance to Irrelevant Alternatives (IIA) Thorough Weak Pareto Optimality (WPO) / Pareto Optimality (PO) Benevolent Symmetry (SYM)Impartial Concavity (CONC)Uses foresight
WPO+SYM+IIA+CONC ⇔ ◦ The Egalitarian Solution or The Utilitarian Quasi- Solution (for a comprehensive problem domain) 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 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
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) PO(S) ⊆ Efficient(S) ⊆ WPO(S) SYMULINIIA 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: = ◦ When ε→0, the first offer is NBS and it is immediately accepted.
= 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... 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. S t+Δt ={x-AΔt|x ∈ S t } We take Δt→0 and t max →∞. 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.
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, where s=∑p i (x). Applying the Cauchy-Schwarz inequality, we get that ∑x i always increases, except when p(x)∝1. Letting t max →∞, 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).
qquestions?