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ECON 3801, Cooperation Games and the market economy

This lecture, a combined session on both Tuesday and Thursday, talks about the different results in cooperative games and their adaptation to the market economy. — Charles Lutwidge Dodgson

Consider three candidates X, Y, Z being voted by three players, and their choices are:

voter 1	voter 2	voter 3	society’s choice
X	Y	Z	x
Y	Z	X	y
Z	X	Y	z

and the collective choice is a tie with the score of 3/2/1 is assigned(?).

the way of putting together people’s choices

non-dictatorship

The (nash) bargaining problem

Think about the example of two farmers, one with corps and one with land. Non-cooperation vs. cooperation (utility pairs)

definition: (F (feasible set), d (non-cooperative pair (d1,d2)))

bargaining solution: a function (a rule) f that makes the solution pair (x1,x2) within F

the feasible set is a convex[^1] set, and the boarder is part of the set.

Axiom R: if (x1,x2) is a pair of points within the feasible set, then (x1,x2) is a bargaining solution Symmetric axiom: if (x1,x2) is feasible, then (x2,x1) must be feasible. Notice the symmetric axis is the diagonal line with a 45-degree angle. cardinality axiom

The bargaining solution must lie on the frontier of the feasible set - to avoid wasteful solutions. To find the bargaining solution, draw the family of rectangular hyperbolas - the indifferent utility curves? - and find the outmost point of the intersection of the curves. In formulas, the nash bargaining solution maximizes: arg max[^2](u1 - d1)*(u2 - d2), which is the shape of the rectangle under the hyperbola.

The max of each player should determine the bargaining solution. Take each of the max points of the two players, the interection defines the kalai-smorodinsky[^3] solution.

[^1] convex: a set is convex if for any two points in the set, the line segment between them is also in the set. Notice that for non-convex sets in the real life, the lottery between solutions is a method to convexing the set. [^2] arg max: the argument that maximizes the function [^3] kalai-smorodinsky: https://en.wikipedia.org/wiki/Kalai%E2%80%93Smorodinsky_bargaining_solution