Pure Competition Game
This is a type of game is the Game Theory where players have exactly opposite interests
- In such games there should be precisely two players (otherwise they couldn’t have the opposite interests)
So a pure competition game is where
- $a \in A, u_1(a) + u_2(a) = c$
- means that if somebody wins, another player loses exactly the amount the first player wins
- this is also called constant sum game
- if $c$ = 0, a game is called a ‘‘zero sum game’’
Zero Sum Games
Matching Pennies
This is a zero-sum game
Rules:
- $p_1$ wants to match, $p_2$ - to mismatch
- each player tosses a coin and record what they have: heads or tails
- if both have the same, $p_1$ wins, $p_2$ looses
- if both have different, $p_1$ looses, $p_2$ wins
Payoff matrix:
| Head | Tail | Head | (1, -1) | (-1, 1) | Tail | (-1, 1) | (1, -1) |
In this game there’s no Nash Equilibrium:
- if $p_2$ knows that $p_1$ plays $H$ he will play $H$
- then if $p_1$ knows that $p_2$ plays $H$, he will play $T$
- so there’s always an incentive to deviate to other alternative
Rock Paper Scissors
Is a generalization of Matching Pennies to 3 alternatives
| Rock | Paper | Scissors | Rock | (0, 0) | (-1, 1) | (1, -1) | Paper | (1, -1) | (0, 0) | (-1, 1) | Scissors | (-1, 1) | (1, -1) | (0, 0) |