Analyzing Zero-Sum Games:
Strategies and Payoffs
Zero-Sum Games
Even or Odd
In this game, players 1 and 2 simultaneously call out one of the two numbers
"one" or "two." Player 1's name is Odd, and he wins if the sum of the
numbers is odd. Player 2's name is Even, and she wins if the sum of the
numbers is even.
The strategy spaces are S1 = S2 = {1, 2}, and the payoffs are given by the
following table:
| \ | 1 | 2 | |---|---|---| | 1 | -2, 2 | 3, -3 | | 2 | 3, -3 | -4, 4 |
(a) The maxmin strategies are σ1 = (3/5, 2/5) for player 1 and σ2 = (7/12,
5/12) for player 2. (b) With the strategy σ1 = (3/5, 2/5), player 1 can
guarantee a nonnegative expected payoff. (c) With the strategy σ1 = (7/12,
5/12), player 1 can guarantee an expected payoff of 1/12. (d) The maxmin
strategy for player 2 is σ2 = (7/12, 5/12), and she can guarantee an
expected payoff of -1/12. (e) The value of the game is v = v1 = 1/12 > -1/12
= -v = v2, so player 1 is advantaged in the game.
Even or Odd Revisited
In this version of the game, the loser pays the winner the product, rather
than the sum, of the numbers called (who wins still depends on the sum).
(a) The payoff matrix is:
| \ | 1 | 2 | |---|---|---| | 1 | -1 | 2 | | 2 | 2 | -4 |
(b) The maxmin strategies are σ1 = σ2 = (2/3, 1/3), and the value of the
game is zero, making it a fair game.
Holmes and Moriarty
Sherlock Holmes boards a train from London to Dover to escape from
Professor Moriarty, who can catch him at Dover. However, Holmes may
detrain at the intermediate station of Canterbury to avoid Moriarty.
The payoffs to Moriarty (and the negatives to Holmes) are given by the
following matrix:
| Moriarty\Holmes | Canterbury | Dover | |-----------------|------------|-------| |
Canterbury | 100 | -50 | | Dover | 0 | 100 |
