S 485/680 Knowledge-Based Agents Game heory Week 7, Lecture 7 What is game theory? Game theory is a formal way to analyze strategic interaction among a group of rational players (or agents) who behave strategically In addition to our own examples, examples and some slides come from various lecture notes available online, including: ) Andrew Moore (MU) 2) an Neill (MU) 3) X. Liu (MU) Game theory has applications Economics Politics etc. 2 What is game theory? We focus on games where: here are at least two rational players Each player has more than one choice he outcome depends on the strategies chosen by all players; there is strategic interaction Example: Six people go to a restaurant. Each person pays his/her own meal a simple decision problem Before the meal, every person agrees to split the bill evenly among them a game 3 Rational decisions A decision maker is assumed to have a fixed range of alternatives to choose from, and his choice influences the outcome of the situation. Each possible outcome is associated with a real number its utility. his can be subjective (how much the outcome is desired) or objective (how good the outcome actually is for the player). In any case, the basic assumption of game theory is: A rational player will make the decision that maximizes his expected utility. 4 ypes of decision situation ecision making under certainty: would you prefer to be paid $00 or punched in the nose? onsequences (A) of each action A are known. A rational agent chooses the action with the highest utility U((A)). or most people, U(paid $00)>U(punched in nose) so they would choose the former. Note that we are only considering the rationality of actions, not preferences: a person who prefers a punch in the nose can still be rational under our definition! 5 ypes of decision situation (2) ecision making under risk: would you wager $00 on the flip of a fair coin? or each action, a probability distribution over possible consequences P( A) is known. A rational agent chooses the action with highest expected utility, P ( A) U( ) or most people, ½ U(gain $00) + ½ U(lose $00) < 0, so they would not take Money this is wager. not utility, since most people are risk-averse! 6
ypes of decision situation (3) ypes of decision situation (4) ecision making under uncertainty: would you rather go to the movies or to the beach? Agents are assumed to have a subjective probability distribution over possible states of nature P(S). he consequence of an action is assumed to be a deterministic function (A, S) of the action A and the state S. A rational agent chooses the action with the highest subjective expected utility, S P )) ecision making under uncertainty: would you rather go to the movies or to the beach? I believe there is a 40% chance that it will rain. I will enjoy a movie whether it rains or not: U((movie, sun) = U((movie, rain)) =. I will not enjoy the beach if it is rainy, but I will have a great time if it is sunny: U((beach, rain)) = -, U((beach, sun)) = 2. SEU(movie) =. SEU(beach) =.4(-)+.6(2) =.8. I m going to the movies! ( S ) U ( ( A, S 7 8 lassic Example: Prisoners ilemma wo suspects held in separate cells are charged with a major crime. owever, there is not enough evidence. Both suspects are told the following policy: If neither confesses then both will be convicted of a minor offense and sentenced to one month in jail. If both confess then both will be sentenced to jail for six months. If one confesses but the other does not, then the confessor will be released but the other will be sentenced to jail for nine months. Prisoner 2 ooperate onfess ooperate -, - -9, 0 Prisoner onfess 9 Example: he battle of the sexes At the separate workplaces, hris and Pat must choose to attend either an opera or a prize fight in the evening. Both hris and Pat know the following: Both would like to spend the evening together. But hris prefers the opera. Pat prefers the prize fight. Pat Opera Prize ight Opera hris 2, 0, 0 Prize ight 0, 0, 2 0 Example: Matching pennies Each of the two players has a penny. wo players must simultaneously choose whether to show the ead or the ail. Both players know the following rules: If two pennies match (both heads or both tails) then player 2 wins player s penny. Otherwise, player wins player 2 s penny. ead ail ead ail -,, -, - -, Static (or simultaneous-move) games of complete information A static (or simultaneous-move) game consists of: A set of players (at least two players) or each player, a set of strategies/actions Payoffs received by each player for the combinations of the strategies, or for each player, preferences over the combinations of the strategies {,,... Player n} S S 2... S n u i (s, s 2,...s n ), for all s S, s 2 S 2,... s n S n. 2 2
Static (or simultaneous-move) games of complete information Simultaneous-move Each player chooses his/her strategy without knowledge of others choices. omplete information Each player s strategies and payoff function are common knowledge among all the players. Assumptions on the players Rationality Players aim to maximize their payoffs Players are perfect calculators Static (or simultaneous-move) games of complete information he players cooperate? No. We focus only on non-cooperative games he timing Each player i chooses his/her strategy s i without knowledge of others choices. hen each player i receives his/her payoff u i (s, s 2,..., s n ). he game ends. Each player knows that other players are rational 3 4 efinition: normal-form or strategic-form representation he normal-form (or strategic-form) representation of a game G specifies: A finite set of players {, 2,..., n}, players strategy spaces S S 2... S n and their payoff functions u u 2... u n where u i : S S 2... S n R. 5 Normal-form representation: 2-player game Matrix representation 2 players: and Each player has a finite number of strategies Example: S ={s, s 2, s 3 } S 2 ={s 2, s 22 } Player s 2 s u (s,s 2 ), u 2 (s,s 2 ) s 2 s 3 u (s 2,s 2 ), u 2 (s 2,s 2 ) u (s 3,s 2 ), u 2 (s 3,s 2 ) s 22 u (s,s 22 ), u 2 (s,s 22 ) u (s 2,s 22 ), u 2 (s 2,s 22 ) u (s 3,s 22 ), u 2 (s 3,s 22 ) 6 Zero sum games Zero sum example Examined in depth by Von Neumann and Morgenstern in the 920s-940s. Most important result: the minimax theorem, which states that under common assumptions of rationality, each player will make the choice that maximizes his minimum expected utility. his choice may be a pure strategy (always making the same choice) or a mixed strategy Payoffs to P/P2 chooses A chooses B chooses a -7 / 7 2 / -2 chooses b -4 / 4 - / (a random choice between pure strategies). 7 8 he value of the game is -! Solution: P always prefers B; P2 (knowing this) prefers Bb to Ba. 3
Non-zero sum games Non-zero sum example an be cooperative (players can make enforceable agreements: I ll cooperate if you do ) or non-cooperative (no prior agreements can be enforced). In non-cooperative games, an agreement must be self-enforcing, in that players have no incentive to deviate from it. Most important concept: Nash Equilibrium. A combination of strategy choices such that no player can increase his utility by changing strategies. Nash s hm: every game has at least one NE. 9 Payoffs to P/P2 chooses A chooses B chooses a 4 / 4 2 / 0 chooses b 0 / 2 / Are there any other Nash equilibria for this game? Aa is a Nash equilibrium: each player gets 4 at Aa, but only 2 if he plays B instead! 20 lassic example: Prisoners ilemma: normal-form representation Set of players: {Prisoner, Prisoner 2} Sets of strategies: S = S 2 = {ooperate, onfess} Payoff functions: u (M, M)=-, u (M, )=-9, u (, M)=0, u (, )=-6; u 2 (M, M)=-, u 2 (M, )=0, u 2 (, M)=-9, u 2 (, )=-6 Players Prisoner Strategies ooperate -, - onfess Payoffs Prisoner 2 ooperate onfess -9, 0 2 Example: he battle of the sexes hris Opera Prize ight Pat Opera Prize ight 2, 0, 0 0, 0, 2 Normal (or strategic) form representation: Set of players: { hris, Pat } (={, }) Sets of strategies: S = S 2 = { Opera, Prize ight} Payoff functions: u (O, O)=2, u (O, )=0, u (, O)=0, u (, O)=; u 2 (O, O)=, u 2 (O, )=0, u 2 (, O)=0, u 2 (, )=2 22 Example: Matching pennies ead ail ead ail -,, -, - -, Normal (or strategic) form representation: Set of players: {, } Sets of strategies: S = S 2 = { ead, ail } Payoff functions: u (, )=-, u (, )=, u (, )=, u (, )=-; u 2 (, )=, u 2 (, )=-, u 2 (, )=-, u 2 (, )= 23 Solving Prisoners ilemma onfess always does better whatever the other player chooses ominated strategy here exists another strategy which always does better regardless of other players choices Players Prisoner Strategies Prisoner 2 ooperate onfess oopoerate -, - onfess Payoffs -9, 0 24 4
efinition: strictly dominated strategy In the normal-form game {S, S 2,..., S n, u, u 2,..., u n }, let s i ', s i " S i be feasible strategies for player i. Strategy s i ' is strictly dominated by strategy s i " if u i (s, s 2,... s i-, s i ', s i+,..., s n ) < u i (s, s 2,... s i-, s i ", s i+,..., s n ) for all s S, s 2 S 2,..., s i- S i-, s i+ S i+,..., s n S n. s i is strictly better than s i efinition: weakly dominated strategy In the normal-form game {S, S 2,..., S n, u, u 2,..., u n }, let s i ', s i " S i be feasible strategies for player i. Strategy s i ' is weakly dominated by strategy s i " if u i (s, s 2,... s i-, s i ', s i+,..., s n ) (but not always =) u i (s, s 2,... s i-, s i ", s i+,..., s n ) for all s S, s 2 S 2,..., s i- S i-, s i+ S i+,..., s n S n. s i is at least as good as s i regardless of other players choices Prisoner ooperate -, - onfess Prisoner 2 ooperate onfess -9, 0 regardless of other players choices U B L R, 2, 0 0, 2 2, 2 25 26 Strictly and weakly dominated strategy A rational player never chooses a strictly dominated strategy. ence, any strictly dominated strategy can be eliminated. A rational player may choose a weakly dominated strategy. 27 PLAYER A -, - Strict omination I S A OL, RUEL WORL. GE OVER I. Player A PLAYER B -9, 0 Assuming B plays, what should I do? Assuming B plays, what oh what should I do? If one of a player s strategies is never the right thing to do, no matter what the opponents do, then it is Strictly ominated 28 Understanding a Game Understanding a Game undamental assumption of game theory: Get Rid of the Strictly ominated strategies. hey Won t appen. undamental assumption of game theory: Get Rid of the Strictly ominated strategies. hey Won t appen. -, - -9, 0 -, - -9, 0 In some cases (e.g. prisoner s dilemma) this means, if players are rational we can predict the outcome of the game. In some cases (e.g. prisoner s dilemma) this means, if players are rational we can predict the outcome of the game. 29 30 5
Understanding a Game undamental assumption of game theory: Get Rid of the Strictly ominated strategies. hey Won t appen. Understanding a Game undamental assumption of game theory: Get Rid of the Strictly ominated strategies. hey Won t appen. -, - -9, 0 -, - -9, 0 In some cases (e.g. prisoner s dilemma) this means, if players are rational we can predict the outcome of the game. In some cases (e.g. prisoner s dilemma) this means, if players are rational we can predict the outcome of the game. 3 32 Understanding a Game undamental assumption of game theory: Get Rid of the Strictly ominated strategies. hey Won t appen. -, - -9, 0 In some cases (e.g. prisoner s dilemma) this means, if players are rational we can predict the outcome of the game. 33 Iterated elimination of strictly dominated strategies If a strategy is strictly dominated, eliminate it he size and complexity of the game is reduced Eliminate any strictly dominated strategies from the reduced game ontinue doing so successively 34 Iterated elimination of strictly dominated strategies: an example Up own Left, 0 0, 3 Middle, 2 0, Right 0, 2, 0 Example: ourists & Natives Only two bars (bar, bar 2) in a city an charge price of $2, $4, or $5 6000 tourists pick a bar randomly 4000 natives select the lowest price bar Up own Left Middle, 0, 2 0, 3 0, 35 Example : Both charge $2 each gets 5,000 customers and $0,000 Example 2: Bar charges $4, Bar 2 charges $5 Bar gets 3000+4000=7,000 customers and $28,000 Bar 2 gets 3000 customers and $5,000 36 6
Example: ourists & Natives Bar $2 $4 $5 $2 0, 0 2, 4 5, 4 Bar 2 $4 4, 2 20, 20 5, 28 $5 4, 5 28, 5 25, 25 Payoffs are in thousands of dollars Example wo firms, Reynolds and Philip, share some market Each firm earns $60 million from its customers if neither do advertising Advertising costs a firm $20 million Advertising captures $30 million from competitor Bar $4 $5 $4 20, 20 5, 28 Bar 2 $5 28, 5 25, 25 37 irm No Ad Ad irm 2 No Ad Ad 60, 60 30, 70 70, 30 40 38 2-player game with finite strategies Strict omination Removal Example S ={s, s 2, s 3 } S 2 ={s 2, s 22 } s is strictly dominated by s 2 if u (s,s 2 )<u (s 2,s 2 ) and u (s,s 22 )<u (s 2,s 22 ). s 2 is strictly dominated by s 22 if u 2 (s i,s 2 ) < u 2 (s i,s 22 ), for i =, 2, 3 Player A I II III IV I 3, 5, 3 2, 3 3, 8 Player B II III 4, 5, 9 5, 8 9, 7 8, 4 6, 2 3, 2, 3 IV 2, 6 9, 3 6, 3 4, 5 Player s u (s,s 2 ), u 2 (s,s 2 ) u (s,s 22 ), u 2 (s,s 22 ) s 2 s 2 u (s 2,s 2 ), u 2 (s 2,s 2 ) u (s,s 2 22 ), u 2 (s 2,s 22 ) s 3 u (s 3,s 2 ), u 2 (s 3,s 2 ) u (s 3,s 22 ), u 2 (s 3,s 22 ) s 22 So is strict domination the best tool for predicting what will transpire in a game? 39 40 Strict omination doesn t capture the whole picture I II III I 3, 5 II 3, 5 III 5, 3 5, 3 6, 6 What strict domination eliminations can we do? What would you predict the players of this game would do? 4 New solution concept: Nash equilibrium M B L 3, 5 3, 5 he combination of strategies (B, R) has the following property: ANNO do better by choosing a strategy different from B, given that player 2 chooses R. ANNO do better by choosing a strategy different from R, given that player chooses B. R 5, 3 5, 3 6, 6 42 7
New solution concept: Nash equilibrium M B L he combination of strategies (B, R ) has the following property: ANNO do better by choosing a strategy different from B, given that player 2 chooses R. R 3.5, 3.6 Nash Equilibrium: idea Nash equilibrium A set of strategies, one for each player, such that each player s strategy is best for her, given that all other players are playing their equilibrium strategies ANNO do better by choosing a strategy different from R, given that player chooses B. 43 44 S S, S i S = 2 S, Λ S 2 Nash Equilibria n S are a NAS EQUILIBRIUM iff arg max i i 2 Si I b II b III b I a 0 4 4 0 5 3 II a 4 0 0 4 5 3 III a 3 5 3 5 6 6 ( S, S, Λ S, S, S Λ S ) i AN u (III a,iii b ) is a N.E. because u n i i+ n u( Ia, IIIb ) ( ) ( ) IIIa, IIIb = max u IIa, IIIb u ( ) IIIa, IIIb u2( IIIa, Ib ) ( III, III ) = max u ( III, II ) u 2 a b 2 a b ( ) u3 IIIa, IIIb 45 If (S *, S 2 *) is an N.E. then player won t want to change their play given player 2 is doing S 2 * If (S *, S 2 *) is an N.E. then player 2 won t want to change their play given player is doing S * ind the NEs: - - 0-9 -9 0-6 -6 0 4 4 0 3 5 4 0 0 4 3 5 Is there always at least one NE? an there be more than one NE? 5 3 5 3 6 6 46 efinition: Nash Equilibrium In the normal-form game {S, S 2,..., S n, u, u 2,..., * * u n }, a combination of strategies ( s,..., sn) is a Nash equilibrium if, for every player i, Given others * * * * * ui ( s,..., si, si, si+,..., sn) choices, player i cannot be betteroff if she deviates * * * * ui ( s,..., si, si, si+,..., sn) for all si Si. hat is, s i * from s i * solves * * * * Maximize u i ( s,..., si, si, si+,..., sn) Subject to si Si Prisoner 2 ooperate onfess ooperate -, - -9, 0 Prisoner onfess 47 2-player game with finite strategies S ={s, s 2, s 3 } S 2 ={s 2, s 22 } (s, s 2 )is a Nash equilibrium if u (s,s 2 ) u (s 2,s 2 ), u (s,s 2 ) u (s 3,s 2 ) and u 2 (s,s 2 ) u 2 (s,s 22 ). s u (s,s 2 22 ), u 2 (s 2,s 22 ) u (s,s 2 ), u 2 (s,s 2 ) u (s,s 22 ), u 2 (s,s 22 ) s 2 s 2 u (s 2,s 2 ), u 2 (s 2,s 2 ) s 3 u (s 3,s 2 ), u 2 (s 3,s 2 ) u (s 3,s 22 ), u 2 (s 3,s 22 ) s 22 48 8
inding a Nash equilibrium: cell-by-cell inspection Up own Up own Left, 0 0, 3, 0 0, 3 Middle, 2 0, Left Middle, 2 0, Right 0, 2, 0 omputing Nash equilibria ere s a neat little trick for finding pure strategy NE in two player games: or each column, color the box(es) with maximum payoff to P red. or each row, color the box(es) with maximum payoff to P2 blue. he Nash equilibria are the set of squares colored both red and blue (purple in our picture). a b c A 0 / 4 4 / 0 5 / 3 B 4 / 0 0 / 4 5 / 3 3 / 5 3 / 5 6 / 6 49 50 Best response function: example M B L R 3.5, 3.6 If chooses L then s best strategy is M If chooses then s best strategy is If chooses R then s best strategy is B If chooses B then s best strategy is R Best response: the best strategy one player can play, given the strategies chosen by all other players 5 Example: ourists & Natives Bar $2 $4 $5 $2 0, 0 2, 4 5, 4 Bar 2 $4 4, 2 20, 20 5, 28 $5 4, 5 28, 5 25, 25 Payoffs are in thousands of dollars what is Bar s best response to Bar 2 s strategy of $2, $4 or $5? what is Bar 2 s best response to Bar s strategy of $2, $4 or $5? 52 Player 2-player game with finite strategies S ={s, s 2, s 3 } S 2 ={s 2, s 22 } s strategy s is her best response to s strategy s 2 if u (s,s 2 ) u (s 2,s 2 ) and u (s,s 2 ) u (s 3,s 2 ). s u (s,s 2 ), u 2 (s,s 2 ) u (s,s 22 ), u 2 (s,s 22 ) s 2 s 2 u (s 2,s 2 ), u 2 (s 2,s 2 ) u (s,s 2 22 ), u 2 (s 2,s 22 ) s 3 u (s 3,s 2 ), u 2 (s 3,s 2 ) u (s 3,s 22 ), u 2 (s 3,s 22 ) s 22 53 Using best response function to find Nash equilibrium In a 2-player game, ( s, s 2 ) is a Nash equilibrium if and only if player s strategy s is her best response to player 2 s strategy s 2, and player 2 s strategy s 2 is her best response to player s strategy s. Prisoner ooperate -, - onfess Prisoner 2 ooperate onfess -9, 0 54 9
Using best response function to find Nash equilibrium: example M B L R 3.5, 3.6 M is s best response to s strategy L is s best response to s strategy B is s best response to s strategy R Example: ourists & Natives Bar $2 $4 $5 $2 0, 0 2, 4 5, 4 Bar 2 $4 4, 2 20, 20 5, 28 $5 4, 5 28, 5 25, 25 Payoffs are in thousands of dollars Use best response function to find the Nash equilibrium. L is s best response to s strategy is s best response to s strategy M R is s best response to s strategy B 55 56 Example: he battle of the sexes Pat Opera Prize ight Opera hris 2, 0, 0 Prize ight 0, 0, 2 Example: Matching pennies ead ail ead -,, - ail, - -, Opera is s best response to s strategy Opera Opera is s best response to s strategy Opera ence, (Opera, Opera) is a Nash equilibrium ight is s best response to s strategy ight ight is s best response to s strategy ight ence, (ight, ight) is a Nash equilibrium 57 ead is s best response to s strategy ail ail is s best response to s strategy ail ail is s best response to s strategy ead ead is s best response to s strategy ead ence, NO Nash equilibrium 58 efinition: best response function In the normal-form game {S, S 2,..., S n, u, u 2,..., u n }, if player, 2,..., i-, i+,..., n choose strategies s,..., si, si+,..., sn, respectively, then player i's best response function is defined by Bi ( s,..., si, si+,..., sn ) = { si Si : ui ( s,..., si, si, si+,..., sn) u ( s,..., s, s, s,..., s ), for all s S } Player i s best response i i i i+ n i i Given the strategies chosen by other players efinition: best response function An alternative definition: Player i's strategy s i Bi ( s,..., si, si+,... sn) if and only if it solves (or it is an optimal solution to) Maximize u i( s,..., si, si, si+,..., sn) Subject to si Si where s,..., si, si+,..., sn are given. Player i s best response to other players strategies is an optimal solution to 59 60 0
Using best response function to define Nash equilibrium Nash equilibrium survive iterated elimination of strictly dominated strategies In the normal-form game {S,..., S n, u,..., u n }, * * a combination of strategies ( s,..., sn ) is a Nash equilibrium if for every player i, * * * B ( s,..., s, s,..., s ) * * s i i i i+ n Up own Left, 0 0, 3 Middle, 2 0, Right 0, 2, 0 A set of strategies, one for each player, such that each player s strategy is best for her, given that all other players are playing their strategies, or A stable situation that no player would like to deviate if others stick to it 6 Up own Left Middle, 0, 2 0, 3 0, 62 he strategies that survive iterated elimination of strictly dominated strategies are not necessarily all Nash equilibrium strategies Example with no NEs among the pure strategies: M B L R 3.5, 3.6 S S 2 S S 2 63 64 Example with no NEs among the pure strategies: S S 2 S 0 0 S 2 0 0 Matching pennies revisited ead ail ead -,, - ail, - -, ead is s best response to s strategy ail ail is s best response to s strategy ail ail is s best response to s strategy ead ead is s best response to s strategy ead 65 ence, NO Nash equilibrium 66
Solving matching pennies ead ail ead ail -,, -, - -, q -q r -r Randomize your strategies to surprise the rival chooses ead and ail with probabilities r and -r, respectively. chooses ead and ail with probabilities q and -q, respectively. Mixed Strategy: Specifies that an actual move be chosen randomly from the set of pure strategies with some specific probabilities. 67 Solving matching pennies ead ail ead ail -,, -, - -, q -q r -r s expected payoffs If chooses ead, -q+(-q)=-2q If chooses ail, q-(-q)=2q- Expected payoffs -2q 2q- 68 Solving matching pennies Solving matching pennies ead ail s best response B (q): or q<0.5, ead (r=) or q>0.5, ail (r=0) or q=0.5, indifferent (0 r ) ead ail -,, -, - -, q -q /2 r r -r /2 Expected payoffs -2q 2q- q 69 ead ail Expected payoffs ead ail -,, -, - -, q -q 2r- -2r r -r s expected payoffs If chooses ead, r-(-r)=2r- If chooses ail, -r+(-r)=-2r Expected payoffs -2q 2q- 70 Solving matching pennies ead ail Expected payoffs ead ail -,, -, - -, q -q 2r- -2r s best response B 2 (r): or r<0.5, ail (q=0) or r>0.5, ead (q=) or r=0.5, indifferent (0 q ) /2 r r -r /2 Expected payoffs -2q 2q- q 7 Solving matching pennies s best response B (q): or q<0.5, ead (r=) or q>0.5, ail (r=0) or q=0.5, indifferent (0 r ) s best response B 2 (r): or r<0.5, ail (q=0) or r>0.5, ead (q=) or r=0.5, indifferent (0 q ) heck r = 0.5 B (0.5) q = 0.5 B 2 (0.5) Player ead ail /2 ead -, ail, - r, - -, -r q -q r /2 Mixed strategy Nash equilibrium q 72 2
Mixed strategy equilibrium Mixed Strategy: A mixed strategy of a player is a probability distribution over the player s strategies. Mixed strategy Nash equilibrium A probability distribution for each player he distributions are mutual best responses to one another in the sense of 2-player mixed strategy Nash Equilibrium he pair of mixed strategies (M A, M B ) are a Nash Equilibrium iff M A is player A s best mixed strategy response to M B AN M B is player B s best mixed strategy response to M A expected payoffs 73 74 undamental heorems In the n-player pure strategy game G={S S 2 S n ; u u 2 u n }, if iterated elimination of strictly dominated strategies eliminates all but the strategies (S *, S 2 * S n *) then these strategies are the unique NE of the game Any NE will survive iterated elimination of strictly dominated strategies [Nash, 950] If n is finite and S i is finite i, then there exists at least one NE (possibly 75 involving mixed strategies) Player Mixed strategy equilibrium: 2- player each with two strategies s ( r ) s 2 (- r ) s 2 ( q ) u (s, s 2 ), u 2 (s, s 2 ) u (s 2, s 2 ), u 2 (s 2, s 2 ) s 22 ( - q ) u (s, s 22 ), u 2 (s, s 22 ) u (s 2, s 22 ), u 2 (s 2, s 22 ) heorem Let ((r*, -r*), (q*, -q*)) be a pair of mixed strategies, where 0 <r*<, 0<q*<. hen ((r*, -r*), (q*, -q*)) is a mixed strategy Nash equilibrium if and only if EU (s, (q*, -q*)) = EU (s 2, (q*, -q*)) EU 2 (s 2, (r*, -r*)) = EU 2 (s 22, (r*, -r*)) hat is, each player is indifferent between her two strategies. 76 Use indifference to find mixed Nash equilibrium (2-player each with 2 strategies) Use heorem 2 to find mixed strategy Nash equilibria Solve EU (s, (q*, -q*)) = EU (s 2, (q*, -q*)) Solve EU 2 (s 2, (r*, -r*)) = EU 2 (s 22, (r*, -r*)) Use Indifference heorem to find mixed strategy Nash equilibrium Matching pennies ( r ) ( r ) ( q ) ( q ) -,, -, - -, is indifferent between playing ead and ail EU (, (q, q)) = q (-) + ( q) = 2q EU (, (q, q)) = q + ( q) (-)=2q 77 EU (, (q, q)) = EU (, (q, q)) 2q = 2q 4q = 2 his give us q = /2 78 3
Use Indifference heorem to find mixed strategy Nash equilibrium Matching pennies ( r ) ( r ) is indifferent between playing ead and ail. EU 2 (, (r, r)) = r +( r) (-) =2r EU 2 (, (r, r)) = r (-)+( r) = 2r EU 2 (, (r, r)) = EU 2 (, (r, r)) 2r = 2r 4r = 2 his give us r = /2 ( q ) ( q ) -,, -, - -, ence, ((0.5, 0.5), (0.5, 0.5)) is a mixed strategy Nash equilibrium by Indifference heorem. 79 Use Indifference heorem to find mixed strategy Nash equilibrium Battle of sexes hris Opera ( r ) Prize ight (-r) Opera (q) 2, 0, 0 Pat Use indifference to find a mixed Nash equilibrium Prize ight (-q) 0, 0, 2 80 Battle of sexes Battle of sexes Pat Pat hris Opera ( r ) Prize ight (-r) Opera (q) 2, 0, 0 Prize ight (-q) 0, 0, 2 hris Opera ( r ) Prize ight (-r) Opera (q) 2, 0, 0 Prize ight (-q) 0, 0, 2 hris expected payoff of playing Opera: 2q hris expected payoff of playing Prize ight: -q hris best response B (q): Prize ight (r=0) if q</3 Opera (r=) if q>/3 Any mixed strategy (0 r ) if q=/3 8 Pat s expected payoff of playing Opera: r Pat s expected payoff of playing Prize ight: 2(-r) Pat s best response B 2 (r): Prize ight (q=0) if r<2/3 Opera (q=) if r>2/3 Any mixed strategy (0 q ) if r=2/3, 82 hris best response B (q): Prize ight (r=0) if q</3 Opera (r=) if q>/3 Any mixed strategy (0 r ) if q=/3 Pat s best response B 2 (r): Prize ight (q=0) if r<2/3 Opera (q=) if r>2/3 Any mixed strategy (0 q ) if r=2/3 Battle of sexes hree Nash equilibria: ((, 0), (, 0)) ((0, ), (0, )) ((2/3, /3), (/3, 2/3)) 2/3 r /3 q 83 Another mixed strategy Nash Equilibrium Example What about this game? Aa and Bb are both pure strategy NE. Are there any mixed NE? Assume there is a mixed strategy NE with P (A) = x, and P 2 (a) = y. A B a 4 / 4 2 / 0 or P to be indifferent between A and B: 4y + 0(-y) = 2y + (-y) y = ⅓. or P2 to be indifferent between a and b: 4x + 0(-x) = 3x + (-x) x = ½. Mixed strategy NE: ((½ A + ½ B), (⅓ a + ⅔ b)). b 0 / 3 / 84 4
Example:Market entry game wo firms, irm and irm 2, must decide whether to put one of their restaurants in a shopping mall simultaneously. Each has two strategies: Enter, Not Enter If either firm plays Not Enter, it earns 0 profit If one plays Enter and the other plays Not Enter then the firm plays Enter earns $500K If both plays Enter then both lose $00K because the demand is limited 85 irm Example: Market entry game Enter ( r ) Not Enter ( r ) irm 2 Enter ( q ) Not Enter ( q ) -00, -00 500, 0 0, 500 0, 0 ow many Nash equilibria can you find? wo pure strategy Nash equilibrium (Not Enter, Enter) and (Enter, Not Enter) One mixed strategy Nash equilibrium ((5/6, /6), (5/6, /6)) hat is r=5/6 and q=5/6 86 Another Example ( r ) L ( q ), R ( q ), 2 B ( r ) 2, 3 0, ynamic Games ow many Nash equilibria can you find? wo pure strategy Nash equilibrium (B, L) and (, R) One mixed strategy Nash equilibrium ((2/3, /3), (/2, /2)) hat is r=2/3 and q=/2 87 88 Entry game An incumbent monopolist faces the possibility of entry by a challenger. he challenger may choose to enter or stay out. If the challenger enters, the incumbent can choose either to accommodate or to fight. he payoffs are common knowledge. Incumbent A In hallenger 2, 0, 0 Out, 2 he first number is the payoff of the challenger. he second number is the payoff of the incumbent. 89 Sequential-move matching pennies Each of the two players has a penny. first chooses whether to show the ead or the ail. After observing player s choice, player 2 chooses to show ead or ail Both players know the following rules: If two pennies match (both heads or both tails) then player 2 wins player s penny. Otherwise, player wins player 2 s penny. -,, -, - -, 90 5
ynamic (or sequential-move) games of complete information A set of players Who moves when and what action choices are available? What do players know when they move? Players payoffs are determined by their choices. All these are common knowledge among the players. 9 efinition: extensive-form representation he extensive-form representation of a game specifies: the players in the game when each player has the move what each player can do at each of his or her opportunities to move what each player knows at each of his or her opportunities to move the payoff received by each player for each combination of moves that could be chosen by the players 92 ynamic games of complete and perfect information Perfect information All previous moves are observed before the next move is chosen. A player knows Who has moved What before she makes a decision 93 A game tree has a set of nodes and a set of edges such that each edge connects two nodes (these two nodes are said to be adjacent) for any pair of nodes, there is a unique path that connects these two nodes Game tree a path from x 0 to x 4 an edge connecting nodes x and x 5 x 0 x x 2 x 3 x 4 x 5 x 6 x 7 x 8 a node 94 Game tree A path is a sequence of distinct nodes y, y 2, y 3,..., y n-, y n such that y i and y i+ are adjacent, for i=, 2,..., n-. We say that this path is from y to y n. We can also use the sequence of edges induced by these nodes to denote the path. he length of a path is the number of edges contained in the path. Example : x 0, x 2, x 3, x 7 is a path of length 3. Example 2: x 4, x, x 0, x 2, x 6 is a path of length 4 a path from x 0 to x 4 x 0 x x 2 x 3 x 4 x 5 x 6 x 7 x 8 95 here is a special node x 0 called the root of the tree which is the beginning of the game he nodes adjacent to x 0 are successors of x 0. he successors of x 0 are x, x 2 or any two adjacent nodes, the node that is connected to the root by a longer path is a successor of the other node. Example 3: x 7 is a successor of x 3 because they are adjacent and the path from x 7 to x 0 is longer than the path from x 3 to x 0 Game tree x 0 x x 2 x 3 x 4 x 5 x 6 x 7 x 8 96 6
If a node x is a successor of another node y then y is called a predecessor of x. In a game tree, any node other than the root has a unique predecessor. Any node that has no successor is called a terminal node which is a possible end of the game Example 4: x 4, x 5, x 6, x 7, x 8 are terminal nodes Game tree x 4 x 0 x x 2 x 3 x 5 x 6 x 7 x 8 97 Any node other than a terminal node represents some player. or a node other than a terminal node, the edges that connect it with its successors represent the actions available to the player represented by the node Game tree -,, - 98, - -, A path from the root to a terminal node represents a complete sequence of moves which determines the payoff at the terminal node Game tree -,, -, - -, Strategy A strategy for a player is a complete plan of actions. It specifies a feasible action for the player in every contingency in which the player might be called on to act. 99 00 Entry game hallenger s strategies In Out Incumbent s strategies Accommodate (if challenger plays In) ight (if challenger plays In) Payoffs Normal-form representation hallenger In Out Incumbent Accommodate ight 2, 0, 0, 2, 2 0 Strategy and payoff In a game tree, a strategy for a player is represented by a set of edges. A combination of strategies (sets of edges), one for each player, induce one path from the root to a terminal node, which determines the payoffs of all players 02 7
Sequential-move matching pennies Sequential-move matching pennies s strategies ead ail s strategies if player plays, if player plays if player plays, if player plays if player plays, if player plays if player plays, if player plays heir payoffs Normal-form representation Player -,, - -,, - -,, -, - -, s strategies are denoted by,, and, respectively. 03 04 Nash equilibrium he set of Nash equilibria in a dynamic game of complete information is the set of Nash equilibria of its normal-form. ind Nash equilibrium ow to find the Nash equilibria in a dynamic game of complete information onstruct the normal-form of the dynamic game of complete information ind the Nash equilibria in the normal-form 05 06 Nash equilibria in entry game wo Nash equilibria ( In, Accommodate ) ( Out, ight ) oes the second Nash equilibrium make sense? incredible threats Remove unreasonable Nash equilibrium Subgame perfect Nash equilibrium is a refinement of Nash equilibrium It can rule out unreasonable Nash equilibria or incredible threats Incumbent hallenger In Out Accommodate 2,, 2 ight 0, 0, 2 07 08 8
A subgame of a game tree begins at a nonterminal node and includes all the nodes and edges following the nonterminal node A subgame beginning at a nonterminal node x can be obtained as follows: remove the edge connecting x and its predecessor the connected part containing x is the subgame Subgame -,, - a subgame, - -, 09 Subgame: example E G 3, E 2, 0, 2 0, 0 G 3, G, 2 0, 0, 2 0, 0 0 Subgame-perfect Nash equilibrium A Nash equilibrium of a dynamic game is subgame-perfect if the strategies of the Nash equilibrium constitute a Nash equilibrium in every subgame of the game. Subgame-perfect Nash equilibrium is a Nash equilibrium. Entry game wo Nash equilibria ( In, Accommodate ) is subgame-perfect. ( Out, ight ) is not subgame-perfect because it does not induce a Nash equilibrium in the subgame beginning at Incumbent. Incumbent A In hallenger 2, 0, 0 Out, 2 Incumbent A 2, 0, 0 Accommodate is the Nash equilibrium in this subgame. 2 ind subgame perfect Nash equilibria: backward induction Starting with those smallest subgames hen move backward until the root is reached Incumbent A In hallenger 2, 0, 0 Out, 2 he first number is the payoff of the challenger. he second number is the payoff of the incumbent. 3 ind subgame perfect Nash equilibria: backward induction G E, 2 0, 0 Subgame perfect Nash equilibrium (G, E) plays, and plays G if player 2 plays E plays E if player plays 3, 2, 0 4 9
Existence of subgame-perfect Nash equilibrium Backward induction: illustration Every finite dynamic game of complete and perfect information has a subgameperfect Nash equilibrium that can be found by backward induction. E G 2, 3, 0 0, 2, 3 5 Subgame-perfect Nash equilibrium (, E). player plays ; player 2 plays E if player plays, plays if player plays. 6 Multiple subgame-perfect Nash equilibria: illustration E Multiple subgame-perfect Nash equilibria E G I J K G I J K 0,, 0, 2, 2, 2, 3 0,, 0, 2, 2, 2, 3 Subgame-perfect Nash equilibrium (, K). player plays player 2 plays if player plays, plays if player plays, plays K if player plays E. 7 Subgame-perfect Nash equilibrium (E, K). player plays E; player 2 plays if player plays, plays if player plays, plays K if player plays E. 8 Multiple subgame-perfect Nash equilibria 0, G, 0, Subgame-perfect Nash equilibrium (, IK). player plays ; player 2 plays if player plays, plays I if player plays, plays K if player plays E. I 2, E J 2, 2 K, 3 9 ynamic games of complete and imperfect information Imperfect information A player may not know exactly Who has made What choices when she has an opportunity to make a choice. Example: player 2 makes her choice after player does. needs to make her decision without knowing what player has made. 20 20