Zero-sum Games The Essentials of a Game Extensive Game Matrix Game Dominant Strategies Prudent Strategies Solving the Zero-sum Game The Minimax Theorem

The Essentials of a Game 1. Players: We require at least 2 players (Players choose actions and receive payoffs.) 2. Actions: Player i chooses from a finite set of actions, S = {s 1,s 2,…..,s n }. Player j chooses from a finite set of actions T = {t 1,t 2,……,t m }. 3. Payoffs: We define P i (s,t) as the payoff to player i, if Player i chooses s and player j chooses t. We require that P i (s,t) + P j (s,t) = 0 for all combinations of s and t. 4. Information: What players know (believe) when choosing actions. ZERO-SUM

The Essentials of a Game 4. Information: What players know (believe) when choosing actions. Perfect Information: Players know their own payoffs other player(s) payoffs the history of the game, including other(s) current action* *Actions are sequential (e.g., chess, tic-tac-toe). Common Knowledge

Extensive Game Player 1 chooses a = {1, 2 or 3} Player 2 b = {1 or 2} Player 1 c = {1, 2 or 3} Payoffs = a 2 + b 2 + c 2 if /4 leaves remainder of 0 or 1. -(a 2 + b 2 + c 2 ) if /4 leaves remainder of 2 or 3. Player1’s decision nodes GAME 1. “Square the Diagonal” (Rapoport: 48-9) Player 2’s decision nodes

Extensive Game How should the game be played? Solution: a set of “advisable” strategies, one for each player. Strategy: a complete plan of action for every possible decision node of the game, including nodes that could only be reached by a mistake at an earlier node. Player1‘s advisable Strategy in red GAME Start at the final decision nodes (in red) Backwards-induction

Extensive Game How should the game be played? Solution: a set of “advisable” strategies, one for each player. Strategy: a complete plan of action for every possible decision node of the game, including nodes that could only be reached by a mistake at an earlier node. Player1‘s advisable Strategy in red GAME 1. Player2’s advisable strategy in green Player1’s advisable strategy in red

Extensive Game How should the game be played? If both player’s choose their advisable (prudent) strategies, Player1 will start with 2, Player2 will choose 1, then Player1 will choose 2. The outcome will be 9 for Player1 (-9 for Player2). If a player makes a mistake, or deviates, her payoff will be less GAME

Extensive Game A Clarification: Rapoport (pp ) claims Player 1 has 27 strategies. However, if we consider inconsistent strategies, the actual number of strategies available to Player 1 is 3 7 = An inconsistent strategy includes actions at decision nodes that would not be reached by correct implementation at earlier nodes, i.e., could only be reached by mistake. Since we can think of a strategy as a set of instructions (or program) given to an agent or referee (or machine) to implement, a complete strategy must include instructions for what to do after a mistake is made. This greatly expands the number of strategies available, though the essence of Rapoport’s analysis is correct.

Extensive Game Complete Information: Players know their own payoffs; other player(s) payoffs; history of the game excluding other(s) current action* *Actions are simultaneous GAME Information Sets

Matrix Game T1T2 T1T2 Also called “Normal Form” or “Strategic Game” Solution = {S 22, T 1 } S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33

Dominant Strategies Definition Dominant Strategy: a strategy that is best no matter what the opponent(s) choose(s). T 1 T 2 T S1S2S3S1S2S3 S1S2S3S1S2S3

Dominant Strategies Definition Dominant Strategy: a strategy that is best no matter what the opponent(s) choose(s). T 1 T 2 T 3 Sure Thing Principle: If you have a dominant strategy, use it! S1S2S3S1S2S3 S1S2S3S1S2S3

Prudent Strategies T 1 T 2 T 3 Player 1’s worst payoffs for each strategy are in red S1S2S3S1S2S3 Definitions Prudent Strategy: A prudent strategy for player i maximizes the minimum payoff she can get from playing different strategies. Such a strategy is simply max s min t P(s,t) for player i.

Prudent Strategies T 1 T 2 T 3 Player 2’s worst payoffs for each strategy are in green S1S2S3S1S2S3 Definitions Prudent Strategy: A prudent strategy for player i maximizes the minimum payoff she can get from playing different strategies. Such a strategy is simply max s min t P(s,t) for player i.

Prudent Strategies T 1 T 2 T S1S2S3S1S2S3 Definitions Prudent Strategy: A prudent strategy for player i maximizes the minimum payoff she can get from playing different strategies. Such a strategy is simply max s min t P(s,t) for player i. Saddlepoint: A set of prudent strategies (one for each player), s. t. (s’, t’) is a saddlepoint, iff maxmin = minmax. We call the solution {S 2, T 1 } a saddlepoint

Prudent Strategies S1S2S3S1S2S3 Saddlepoint: A set of prudent strategies (one for each player), s. t. (s’, t’) is a saddlepoint, iff maxmin = minmax.

Mixed Strategies Left Right L R L R Player 1 Player 2 GAME 2: Button-Button Player 1 hides a button in his Left or Right hand. Player 2 observes Player 1’s choice and then picks either Left or Right. Draw the game in matrix form.

Mixed Strategies Left Right L R L R Player 1 Player 2 GAME 2: Button-Button Player 1 has 2 strategies; Player 2 has 4 strategies: LRLR LL RR LR RL

Mixed Strategies Left Right L R L R Player 1 Player 2 GAME 2: Button-Button The game can be solve by backwards-induction. Player 2 will … LRLR LL RR LR RL

Mixed Strategies Left Right L R L R Player 1 Player 2 GAME 2: Button-Button The game can be solve by backwards-induction. … therefore, Player 1 will: LRLR LL RR LR RL

Mixed Strategies Left Right L R L R Player 1 Player L R L R GAME 2: Button-Button What would happen if Player 2 cannot observe Player 1’s choice?

Solving the Zero-sum Game GAME Definition Mixed Strategy: A mixed strategy for player i is a probability distribution over all strategies available to player i. Let (p, 1-p) = prob. Player I chooses L, R. (q, 1-q) = prob. Player 2 chooses L, R. L R LRLR

Solving the Zero-sum Game GAME Then Player 1’s expected payoffs are: EP(L) = -2(p) + 2(1-p) = 2 – 4p EP(R) = 4(p) – 1(1-p) = 5p – 1 L R LRLR (p) (1-p) (q) (1-q) 01 p EP(L) = 2 – 4p EP(R) = 5p – 1 EP p*=1/

Solving the Zero-sum Game GAME Player 2’s expected payoffs are: EP(L) = 2(q) – 4(1-q) = 6q – 4 EP(R) = -2(q) + 1(1-q) = -3q + 1 EP(L) = EP(R) => q* = 5/9 L R LRLR (p) (1-p) (q) (1-q)

Solving the Zero-sum Game Player 1 EP(L) = -2(p) + 2(1-p) = 2 – 4p EP(R) = 4(p) – 1(1-p) = 5p – 1 0 p 1 q -EP 2 p*=1/ /3 = EP 1 * = - EP 2 * =-2/3 This is the Value of the game. EP q*= 5/9 Player 2 EP(L) = 2(q) – 4(1-q) = 6q – 4 EP(R) = -2(q) + 1(1-q) = -3q + 1

Solving the Zero-sum Game GAME Then Player 1’s expected payoffs are: EP(T 1 ) = -2(p) + 2(1-p) EP(T 2 ) = 4(p) – 1(1-p) EP(T 1 ) = EP(T 2 ) => p* = 1/3 And Player 2’s expected payoffs are: (V)alue = 2/3 L R LRLR (p) (1-p) (q) (1-q) (Security) Value: the expected payoff when both (all) players play prudent strategies. Any deviation by an opponent leads to an equal or greater payoff.

The Minimax Theorem Von Neumann (1928) Every zero sum game has a saddlepoint (in pure or mixed strategies), s.t., there exists a unique value, i.e., an outcome of the game where maxmin = minmax.