Review: Proof of Arrow’s Impossibility Theorem

Deferred to textbook, page 224.


Zero Sum Games

Recall the payoff diagram from the first few notes. This will be useful later on:

Player 1 \ Player 2 A B C
A -1 0 2
B -2 3 4
C 1 2 3

A zero-sum game has the property that the sum of the outcomes will be 0.

Looking at outcomes, we have a distribution of values for the game. If the expected value of the game is 0, it is called fair. If the value is positive, it favors Player 1. If the value is negative, it favors Player 2.


Matching Pennies

Player 1 and 2 choose heads or tails, independently. If choices match, Player 2 gets 1 cent from Player 1, otherwise Player 1 gets 1 cent from Player 2.

Payoff diagram (aka strategic description):

P1 \ P2 H T
H -1 1
T 1 -1

Odd-Even

Both players call out 1 or 2. If the sum is even, P2 wins. Otherwise, P1 wins.

P1 \ P2 1 2
1 2 -3
2 -3 4

If Player 2 calls 1, then on average, Player 1 wins 1/2. If Player 2 calls 2, then on average, Player 1 wins -1/2.

Suppose instead Player 1 calls 1 3/5 of the time.

  • If P2 calls 1, the expected value is 0.
  • If P2 calls 2, the expected value is 0.2.

Based on our payoff diagram,

  • Player 1 wants to maximize the minimum of
  • Player 2 wants to minimize the maximum of