UNIT II: The Basic Theory Zero-sum Games Nonzero-sum Games Nash Equilibrium: Properties and Problems Bargaining Games Bargaining and Negotiation Review Midterm3/24 2/16

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

The Essentials of a Game 4. Information: What players know (believe) when choosing actions. Complete Information: Players know their own payoffs other player(s) payoffs the history of the game, excluding other(s) current action* *Actions are simultaneous (e.g., matrix games). 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 Enumerating Strategies: In the extensive form game with perfect information (Game 1), Player 2 has 8 strategies: GAME , 1, 1 1, 1, 2 1, 2, 1 1, 2, 2

Extensive Game Enumerating Strategies: In the extensive form game with perfect information (Game 1), Player 2 has 8 strategies: GAME , 1, 1 1, 1, 2 1, 2, 1 1, 2, 2 2 1

Extensive Game Enumerating Strategies: In the extensive form game with perfect information (Game 1), Player 2 has 8 strategies: GAME , 1, 1 1, 1, 2 1, 2, 1 1, 2, 2 2 1

Extensive Game Enumerating Strategies: In the extensive form game with perfect information (Game 1), Player 2 has 8 strategies: GAME , 1, 1 1, 1, 2 1, 2, 1 1, 2, 2 2 2

Extensive Game Enumerating Strategies: In the extensive form game with perfect information (Game 1), Player 2 has 8 strategies: GAME 1. 1, 1, 1 1, 1, 2 1, 2, 1 1, 2, 2 2, 1, 1 2, 1, 2 2, 2, 1 2, 2,

Extensive Game Enumerating Strategies: In the extensive form game with perfect information (Game 1), Player 2 has 8 strategies: GAME , 1, 1 1, 1, 2 1, 2, 1 1, 2, 2 2, 1, 1 2, 1, 2 2, 2, 1 2, 2, 2 2 1

Extensive Game Enumerating Strategies: In the extensive form game with perfect information (Game 1), Player 2 has 8 strategies: GAME , 1, 1 1, 1, 2 1, 2, 1 1, 2, 2 2, 1, 1 2, 1, 2 2, 2, 1 2, 2, 2 2 1

Extensive Game Enumerating Strategies: In the extensive form game with perfect information (Game 1), Player 2 has 8 strategies: GAME , 1, 1 1, 1, 2 1, 2, 1 1, 2, 2 2, 1, 1 2, 1, 2 2, 2, 1 2, 2, 2 2 2

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

Extensive Game In the extensive form game with complete information, Player 2 has only 2 strategies: GAME , 1, 1 1, 1, 2 1, 2, 1 1, 2, 2 T 1 (Always choose 1)

Extensive Game In the extensive form game with complete information, Player 2 has only 2 strategies: GAME , 1, 1 2, 1, 2 2, 2, 1 2, 2, … and T 2 (Always choose 2)

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

Dominant Strategies 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 S1S2S3S1S2S3 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. Also called, Maximin Strategy

Prudent Strategies T 1 T 2 T 3 Player 1’s worst payoffs for each strategy are in red S1S2S3S1S2S3 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 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 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 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 (Rapoport: ) 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: L or R LRLR LL RR LR RL

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

Mixed Strategies Left Right L R L R Player 1 Player L R L R GAME 2: Button-Button

Solving the Zero-sum Game GAME 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 1 (L|p) = -2(p) + 2(1-p) = 2 – 4p EP 1 (R|p) = 4(p) – 1(1-p) = 5p – 1 L R LRLR (p) (1-p) (q) (1-q) 01 p EP p*=1/ EP 1 (L|p) = 2 – 4p EP 1 (R|p) = 5p – 1

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

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

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.

Nonzero-sum Games Examples: Bargaining Duopoly International Trade

Nonzero-sum Games The Essentials of a Game Eliminating Dominated Strategies Best Response Nash Equilibrium Duopoly: An Application Solving the Game Existence of Nash Equilibrium Properties and Problems See: Gibbons, Game Theory for Applied Economists (1992): 1-51.

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 }. 3A. Payoffs: We define P i (s,t) as the payoff to player i, if Player i chooses s and player j chooses t. We allow that P i (s,t) + P j (s,t) = Information: What players know (believe) when choosing actions. NONZERO-SUM

Eliminating Dominated Strategies 1,0 1,20,1 0,3 0,12,0 1,0 1,2 0,3 0,1 1,0 1,2 L M R TBTB R is strictly dominated by M, so the game can be reduced to Now, B is strictly dominated by T... TBTB T {(T, M)}

Eliminating Dominated Strategies Definition Best Response Strategy: a strategy, s’, is a best response strategy iff P i (s’,t) > P i (s,t) for all s. 0,4 4,0 5,3 4,0 0,4 5,3 3,5 3,5 4,4 S1S2S3S1S2S3 T 1 T 2 T 3 A dominated strategy will never be played by a rational player.

Eliminating Dominated Strategies Definition Best Response Strategy: a strategy, s’, is a best response strategy iff P i (s’,t) > P i (s,t) for all s. 0,4 4,0 5,3 4,0 0,4 5,3 3,5 3,5 4,4 S1S2S3S1S2S3 T 1 T 2 T 3 A strategy can be dominated by a mixture of other strategies … T 3 is dominated by a mixture of T 1 and T 2.

Eliminating Dominated Strategies Definition Best Response Strategy: a strategy, s’, is a best response strategy iff P i (s’,t) > P i (s,t) for all s. 0,4 4,0 5,3 4,0 0,4 5,3 3,5 3,5 4,4 S1S2S3S1S2S3 T 1 T 2 T 3 A strategy can be dominated by a mixture of other strategies … T 3 is dominated by a mixture of T 1 and T 2.

Eliminating Dominated Strategies Definition Best Response Strategy: a strategy, s’, is a best response strategy iff P i (s’,t) > P i (s,t) for all s. 0,4 4,0 5,3 4,0 0,4 5,3 3,5 3,5 4,4 S1S2S3S1S2S3 T 1 T 2 T 3 A strategy can be dominated by a mixture of other strategies … Now S 3 is dominated by a mixture of S 1 and S 2.

Eliminating Dominated Strategies Definition Best Response Strategy: a strategy, s’, is a best response strategy iff P i (s’,t) > P i (s,t) for all s. 0,4 4,0 5,3 4,0 0,4 5,3 3,5 3,5 4,4 S1S2S3S1S2S3 T 1 T 2 T 3 A strategy can be dominated by a mixture of other strategies … Now S 3 is dominated by a mixture of S 1 and S 2.

Eliminating Dominated Strategies Definition Best Response Strategy: a strategy, s’, is a best response strategy iff P i (s’,t) > P i (s,t) for all s ,4 4,0 5,3 4,0 0,4 5,3 3,5 3,5 6,6 S1S2S3S1S2S3 S1S2S3S1S2S3 Does either player have a dominant strategy? A dominated strategy? T 1 T 2 T 3

Best Response Best Response Strategy: a strategy, s*, is a best response strategy, iff the payoff to (s*,t) is at least as great as the payoff to (s,t) for all s ,4 4,0 5,3 4,0 0,4 5,3 3,5 3,5 6,6 S1S2S3S1S2S3 S1S2S3S1S2S3 T 1 T 2 T 3

Best Response Best Response Strategy: a strategy, s*, is a best response strategy, iff the payoff to (s*,t) is at least as great as the payoff to (s,t) for all s ,4 4,0 5,3 4,0 0,4 5,3 3,5 3,5 6,6 S1S2S3S1S2S3 S1S2S3S1S2S3 T 1 T 2 T 3

Nash Equilibrium Best Response Strategy: a strategy, s*, is a best response strategy, iff the payoff to (s*,t) is at least as great as the payoff to (s,t) for all s ,4 4,0 5,3 4,0 0,4 5,3 3,5 3,5 6,6 S1S2S3S1S2S3 S1S2S3S1S2S3 T 1 T 2 T 3 Nash Equilibrium: a set of best response strategies (one for each player), (s*, t*) such that s* is a best response to t* and t* is a b.r. to s*. {(S 3,T 3 )}

Nash Equilibrium ,4 2,3 1,5 3,2 1,1 0,0 5,1 0,0 3,3 S1S2S3S1S2S3 S1S2S3S1S2S3 Nash equilibrium need not be efficient. T 1 T 2 T 3

Nash Equilibrium ,1 0,0 0,0 0,0 1,1 0,0 0,0 0,0 1,1 S1S2S3S1S2S3 S1S2S3S1S2S3 Nash equilibrium need not be unique. A COORDINATION PROBLEM T 1 T 2 T 3

Nash Equilibrium ,1 0,0 0,0 0,0 1,1 0,0 0,0 0,0 1,1 S1S2S3S1S2S3 S1S2S3S1S2S3 Nash equilibrium need not be unique. What is the effect of repeated play? T 1 T 2 T 3

Nash Equilibrium ,1 0,0 0,0 0,0 1,1 0,0 0,0 0,0 3,3 S1S2S3S1S2S3 S1S2S3S1S2S3 Inefficient and multiple Nash equilibrium. T 1 T 2 T 3

Duopoly Consider a market in which two identical firms can produce a good with marginal cost = $2 per unit (assume no fixed cost). Market demand can be described by: P(rice) = 8 – Q(uantity) Where Q is total industry output (Q = q 1 + q 2 )

Duopoly Consider a market in which two identical firms can produce a good with marginal cost = $2 per unit (assume no fixed cost). For each firm, Profit (  ) = Total Revenue – Total Cost = Pq – 2q Each firm will choose a level of output q i, to maximize its profit, taking into account what it expects the other firm to produce.

Duopoly Consider a market in which two identical firms can produce a good with marginal cost = $2 per unit (assume no fixed cost). For each firm, Profit (  ) = Total Revenue – Total Cost = Pq – 2q Each firm will choose a level of output q, to maximize its profit, taking into account what it expects the other firm to produce.

Duopoly Each firm’s profit maximizing level of output, q*, is a function of the other firm’s output. q13 q13 6 q 2 P = 8 - Q Demand Condition  1 = Total Revenue – Total Costq 1 - 2q 1 = 6q 1 - q q 2 q 1 FOC: 6 - 2q 1 - q 2 = 0 q q 1 * = 3 – 1/2q 2 

Duopoly Each firm’s profit maximizing level of output, q*, is a function of the other firm’s output. q13 q13 6 q 2 P = 8 - (q 1 +q 2 )  1 = Pq 1 - 2q 1 = [8 - (q 1 +q 2 )]q 1 - 2q 1 = 6q 1 - q q 2 q 1 = 6 - 2q 1 - q 2 = 0 q q 1 * = 3 – 1/2q 2 

Duopoly Each firm’s profit maximizing level of output, q*, is a function of the other firm’s output. q13 q13 6 q 2 P = 8 - (q 1 +q 2 )  1 = Pq 1 - 2q 1 = [8 - (q 1 +q 2 )]q 1 - 2q 1 = 6q 1 - q q 2 q 1 = 6 - 2q 1 - q 2 = 0 q q 1 * = 3 – 1/2q 2  d1d1 dq1dq1 FOC:

Duopoly A Nash Equilibrium is a pair of “best responses,” such that q 1 * is a best response to q 2 * and q 2 * is a best response to q 1 *. q 1 q 1 * = 2 q 2 * = 2q 2 q 2 * = 3 - 1/2q 1 q 1 * = 3 - 1/2q 2

Duopoly A Nash Equilibrium is a pair of “best responses,” such that q 1 * is a best response to q 2 * and q 2 * is a best response to q 1 *. q1q1* q1q1* q 2 * q 2 Is this the best they can do?

Duopoly A Nash Equilibrium is a pair of “best responses,” such that q 1 * is a best response to q 2 * and q 2 * is a best response to q 1 *. q1q1* q1q1* q 2 * q 2 Is this the best they can do?

Duopoly A Nash Equilibrium is a pair of “best responses,” such that q 1 * is a best response to q 2 * and q 2 * is a best response to q 1 *. q1q1* q1q1* q 2 * q 2 Is this the best they can do? If Firm 1 reduces its output while Firm 2 continues to produce q 2 *, the price rises and Firm 2’s profits increase.

Duopoly A Nash Equilibrium is a pair of “best responses,” such that q 1 * is a best response to q 2 * and q 2 * is a best response to q 1 *. q1q1* q1q1* q 2 * q 2 Is this the best they can do? If Firm 2 reduces its output while Firm 1 continues to produce q 1 *, the price rises and Firm 1’s profits increase.

Duopoly A Nash Equilibrium is a pair of “best responses,” such that q 1 * is a best response to q 2 * and q 2 * is a best response to q 1 *. q1q1* q1q1* q 2 * q 2 Is this the best they can do? There are a range of outcomes to the SW that make both firms better off.

Duopoly Consider a market in which two identical firms can produce a good with marginal cost = $2 per unit (assume no fixed cost). Market demand can be described by: P(rice) = 8 – Q(uantity) Where Q is total industry output (Q = q 1 + q 2 ) Assume that each firm can only choose a discrete quantity 0, 1, 2, 3, 4. What is the matrix form of this game? What is the best response for Firm 1, if it thinks Firm 2 will produce 3 units of output? If it thinks Firm 2 will produce 4 units of output? Find the Nash equilibrium in the market.

Duopoly P = 8 - Q Q = q 1 + q 2  i = (P - 2)q i q 1 q 2 P  1 

Duopoly q 1 q 2 P  1  ,00,5 0,8 0,9 0,8 5,04,4 3,6 2,6 1,4 8,06,3 4,4 2,3 0,0 9,06,2 3,2 0,0 -3,-4 8,04,10,0 -4,-3 -8,

Duopoly q 1 q 2 P  1  q 1 q 1 (q 2 ) q 2 (q 1 ) q2q2 Best Response Functions NE

Duopoly q 1 q 2 P  1  ,00,5 0,8 0,9 0,8 5,04,4 3,6 2,6 1,4 8,06,3 4,4 2,3 0,0 9,06,2 3,2 0,0 -3,-4 8,04,10,0 -4,-3 -8,

Duopoly In a Nash Equilibrium, no firm can increase its profits by changing its output unilaterally. It is strategically stable or self-enforcing agreement. A Nash Equilibrium does not necessarily maximize joint (or industry) profits. But improving the situation creates an enforcement problem. In general, we observe a tension between –Collusion: maximize joint profits –Competition: capture a larger share of the pie

Next Time 2/23Nash Eq: problems & properties Schelling, Strategy of Conflict: