Unformatted text preview: GAME THEORY Lecture 79 Stable Play: Nash Equilibria in discrete games Instructor: Nuno Limão slides_414_7_9_2011_no_solution 1 OUTLINE Nash Equilibria: motivation and definition Nash equilibria: applying best response to solve games Nash Equilibria: Background and justification Selecting among Nash Equilibria Nash and its relationship to alternative types of equilibria N‐player games: motivation N‐player Symmetric Games: Tipping and congestion effects N‐player Asymmetric Games slides_414_7_9_2011_no_solution 2 NASH EQUILIBRIA: MOTIVATION AND DEFINITION Motivation If the game is not dominance solvable then we are left with many (possibly all) strategies and thus no prediction about behavior in the game, Experiment’s results: even without dominant strategy, saw players choosing what turns out to be a stable outcome, or equilibrium (Bottom Right) Left
Right
Top
$1.20, $1.20 $0.00, $1.00
Bottom $1.00, $0.00 $0.30, $0.30 What do we mean by a stable outcome? Suppose that you are column player and you anticipate that the equilibrium involves row playing bottom and you right. Can you do better by deviating from right to left? No, would go from 0.3 to 0 Almost 10% also played Top and Left. Is this stable according to the definition above? How about Bottom left or top right? slides_414_7_9_2011_no_solution 3 Definition: A strategy profile is a Nash equilibrium if each player’s strategy maximizes their payoff, given the strategies used by the other players Note the two important components required for a Nash equilibrium Players are rational (as before) Beliefs are accurate each player must correctly anticipate the equilibrium strategy used by their opponents (much stronger than “reasonable beliefs” condition used before) Formal definition: Consider a game with n players: 1, 2, ... n. Let Si be player i’s strategy set and let si Si be one of player i’s strategies. Let s*‐i represent strategies for all of the players except i. And so s*‐i has n‐1 elements and s*‐i = (s*1, s*2, …. , s*i‐1, s*i+1, …. s*n). A strategy profile (s*i, s*‐i) is a Nash equilibrium if and only if for all i = 1, 2, … n we have Vi(s*i , s*‐i) Vi(si, s*‐i) for all si Si slides_414_7_9_2011_no_solution 4 How can we find the Nash equilibrium in discrete games (those with finite number of strategies for each player)? We must compare payoff for each player’s strategy against all other payoffs from alternative strategies when facing same opponent strategy. Exhaustive search but simplified if we break into steps, i.e. best replies Definition of a best reply (or best response) A best reply for player i to s‐i is a strategy that maximizes i’s payoff given that the other n‐1 players use strategies s‐i. [There is one best‐reply for each possible configuration of other player strategies] Formally, s*i is a best response to s‐i if and only if Vi(s*i, s‐i) Vi(si, s‐i) for all si Si. Then we can say that a strategy profile (s*i, s*‐i) is a Nash equilibrium if s*i is a best response to s*‐i for all i = 1, 2, ……, n. slides_414_7_9_2011_no_solution 5 Applying BR method for row player (underline) If column goes left then best response is top If column goes right then best response is bottom Left
Right
Top
$1.20, $1.20 $0.00, $1.00
Bottom $1.00, $0.00 $0.30, $0.30 Applying BR method for column player (underline) If row goes top then best response is left If row goes bottom then best response is right Left
Right
Top
$1.20, $1.20 $0.00, $1.00
Bottom $1.00, $0.00 $0.30, $0.30 Nash equilibrium: strategies that are best responses for both players so : TL and BR slides_414_7_9_2011_no_solution 6 NASH EQUILIBRIA: APPLYING BEST RESPONSE TO SOLVE GAMES Classic 2x2 examples: Chicken, Prisoner’s Dilemma, Coordination, Stag hunt Game of chicken setup Two drivers head towards each other and each must decide whether to swerve or hang tough. If at least one driver swerves, a collision is avoided. If no one swerves, they die. The first to swerve gets a lower payoff than the second (i.e. is a “chicken”) Exercise Are there any Nash equilibria? Are there other examples that chicken applies to? slides_414_7_9_2011_no_solution 7 Solution to Chicken 2 (pure) Nash Equilibria: (swerve, hang tough) and (hang tough, swerve) If 2 chooses “hang tough” then 1 is better off choosing “swerve.” If 1 chooses “swerve” then 2 is better off choosing “hang tough” Does 1 or 2 swerve? No prediction! Would need additional information about players or additional modeling (e.g. allow for a commitment device such as taking out the wheel) Other examples Cuban missile crisis Movie release dates When competing for a resource where if both fight to “get it” then its value is smaller than if they share but if one of them can easily take it all (hang tough) will be even better. slides_414_7_9_2011_no_solution 8 Prisoner’s Dilemma (recall) Are there any Nash equilibria? Yes. Underline the best response for each. Dilbert Ashok Fink
Mum
(Defect) (Cooperate) Fink
10,10 (Defect) Mum
‐25,0 (Cooperate) 0,‐25 ‐3,‐3 What is the relationship between Nash and dominant strategy equilibrium? Same since the dominant strategy is by definition the best response that a player has against anything the other player may do Moreover, if there is a dominant strategy equilibrium there is also a unique NE slides_414_7_9_2011_no_solution 9 Typology of games and incentives as initial guide to search for NE (e.g. useful in “large” games) Pure conflict (constant‐sum or zero‐sum games) Recall setup in second class of firms splitting market Two players: company L, B choose price ($1 or $2) simultaneously and lower price company takes the full market. Same price leads to even split At $2, demand = 1 million units; at $1, demand = 2 million units Payoffs = Revenue ‐ fixed cost (1 million) ; variable cost is 0 NE (=dominant strategy eq.): Fierce competition L $1 $2
$1 B $2 0,0
1,‐1 ‐1,1
0,0 Is there always a (pure strategy) Nash equilibrium in conflict (or other games)? slides_414_7_9_2011_no_solution 10 Some games have no pure strategy NE, e.g. RPS L B R P S R
P 0,0
1,‐1 ‐1,1 0,0 1,‐1
‐1,1 S ‐1,1 1,‐1 0,0 There still exists a Nash equilibrium but it involves “mixed” (as opposed to “pure”) strategies, where with some probability go R, P or S. [chapter 7] slides_414_7_9_2011_no_solution 11 Another classic: Pure coordination game (recall meet in NY game or Ipad apps development) 2 Grand Central 1,1 0,0 1 Grand central
Empire state Empire state 0,0
1,1 Nash equilibria Two (pure strategy) NE Typical of coordination games these are along diagonal. Thus for larger games use this insight to find NE (at least some initial ones) Can’t predict which of them will emerge. Other examples: conventions such as driving on right or left, adopting operating systems slides_414_7_9_2011_no_solution 12 Combination of conflict and mutual interest elements: Chicken (recall) both drivers want to avoid (tough, tough), so they have a mutual interest. They disagree on ranking (tough, swerve) preferred by 1 and (swerve, tough) by 2 so conflict slides_414_7_9_2011_no_solution 13 Combination of conflict and mutual interest: Battle of the sexes (or calling game) Colleen is chatting with Winnie and suddenly they are disconnected. Should Colleen call Winnie, or wait? If both call then busy, if only one does then must pay the call There are two Nash equilibria: (Call, Wait) and (Wait, Call). slides_414_7_9_2011_no_solution 14 Two common points to note above in Chicken and Battle of the sexes: Both games are symmetric : (1) each player has the same strategy, (2) same payoff associated with a strategy and (3) if we switch the players strategies their payoffs switch as well In symmetric game either both (Swerve, hang tough) and (hang tough, swerve) are NE (as above) or neither are. NE display “anti‐coordination” feature: want to match different strategies. This should be clear from description of the games and by understanding those incentives it will be easier to determine which are the likely NE to begin with. slides_414_7_9_2011_no_solution 15 Combination of mutual interest and conflict: Stag hunt game captures the conflict between gains from cooperation and individual safety 2 starving hunters can choose to hunt stag or hare Stag requires cooperation (bigger, etc) and yields higher payoff for both Hare is easy prey but yields lower payoff then stag but higher than if a player goes after stag and the other hunter defects. It is safer since assured at least 1. 2 Stag Hare 1 stag
hare 2,2 1,0 0,1 1,1 There are 2 pure NE, as should be intuitive from the description of the game Stag, Stag has higher payoffs (is payoff dominant) Hare, hare is risk‐dominant (assured at least a payoff of 1) Several examples in biology and social interactions that feature this trade‐off and generate this type of payoff structure slides_414_7_9_2011_no_solution 16 Using bestreply method in 3player games: matching letters vs. cool new top Each player gets a payoff of 2 if they all wear their lettered T‐shirts (e.g. A,C,E spells a contestant's name) If a player wears non‐lettered t‐shirt but a cool new BEBE top then assured at least 1. Others get zero since they can’t spell anything cool (AC, CA, AE, EA, EC, CE: definitely NOT cool!) Exercise: What are the NE? [hint: think about the incentives, as described in the setup] slides_414_7_9_2011_no_solution 17 Solution: Modified version of stag huntgame so there is a payoff dominant strategy: ACE and a risk dominant strategy: all BEBE slides_414_7_9_2011_no_solution 18 NASH EQUILIBRIA: BACKGROUND AND JUSTIFICATION Nash background and contribution Von Neuman pioneered analysis of 2‐player constant‐sum games in 1920’s and showed the existence of a solution (the minimax theorem) John Nash’s 28 page dissertation contained the basis of what later became known as Nash equilibria and proved its existence in general situations (eventually won him a Nobel in Economics) Two aspects of NE are likely to account for its widespread use in economics (cf. Dixit’s description in “John Nash: Founder of Modern Game Theory”) NE exists in general situations with n‐players and non‐constant sum games that are prevalent in economics NE assumes that each player chooses own strategy to maximize individual payoff while taking other’s strategies as given and an equilibrium is when those strategies are all mutually consistent. Thus it is a natural “strategic” counterpart to Walrasian theory of competitive markets. slides_414_7_9_2011_no_solution 19 Hollywood thought Nash was interesting enough to make a movie “A Beautiful Mind”, but do they understand the concept of a Nash equilibrium? http://www.gametheory.net/media/Beautiful.mov slides_414_7_9_2011_no_solution 20 NE justification NE may be useful in describing possible stable outcomes but requires players’ conjectures about the other’s play to be correct, why may this be reasonable? NE as a necessary condition if there is a unique predicted outcome slides_414_7_9_2011_no_solution 21 In several situations the NE describe only the set of stable outcomes but says nothing about which will actually be picked if more than one exists E.g. who swerves in chicken? or where do we end up in the coordination game? So why focus on NE at all? There are alternative arguments focusing on why one of the outcomes emerges as “obvious” and then appeal to previous argument, e.g. Focal points: if payoffs in coordinating in NY game are much higher at Empire State (great view) than at Grand Central then ES becomes an obvious outcome Pre‐play communication and self‐enforcing agreement: If players can indicate what they will do and both understand that what the other stated is stable then this can help them coordinate strategies and agreement becomes focal. NE as a stable social convention: over time conventions develop (drive on the left OR the right, etc) so the convention becomes focal slides_414_7_9_2011_no_solution 22 SELECTING AMONG NASH EQUILIBRIA Multiplicity of equilibria undermine predictive power of theory so considerable work devoted to refine or select a subset of such equilibria Multiple equilibria in some symmetric games with asymetric equilibria (e.g.: chicken (Swerve, tough) and (tough, swerve)) difficult to select since there is no obvious criteria but provides some guidance (someone will serve and another will hang tough) and the difference in the outcome is then about how the gains are distributed. Multiplicity in other games, e.g. coordination May be more troubling since they predict that two very different outcomes may occur (e.g. adopt one standard or another) so little predictive power May be more amenable to applying certain refinement or selection criteria if payoffs take a certain form, slides_414_7_9_2011_no_solution 23 Multiplicity and refinement in coordination game: e.g. driving convention game for two american drivers 2
Drive right Drive left 5,5 0,0 0,0 2,2 1 Drive right
Drive left 2 NE Driving right equilibrium has higher payoff for each of players than the left equilibrium suggests an obvious selection criterion Selection criterion I (Pareto criterion): Focus on payoff‐dominant Nash equilibrium Find all NE (so no incentive to deviate and consistent w/ individual rationality:) Select the equilibrium for which each player is at least as well off or better slides_414_7_9_2011_no_solution 24 Selection criterion II: Focus on undominated Nash equilibria Find all NE Focus only on equilibria that do not use weakly dominated strategies Exercise: Find the undominated Nash equilibria of the following game slides_414_7_9_2011_no_solution 25 Solution: slides_414_7_9_2011_no_solution 26 Other notes on selection criteria There are other selection criteria, this is an important area of research in GT e.g. in a symmetric game a symmetric equilibirum may be more compelling than an asymmetric one There is disagreement about the merits of different criteria and which to choose, for example it is possible that criterion I and II give conflicting answers (e.g. work through the example below from fig 5.12, p. 140) slides_414_7_9_2011_no_solution 27 NASH AND ITS RELATIONSHIP TO ALTERNATIVE TYPES OF EQUILIBRIA What are the relationships between the main equilibrium concepts studied so far? The set of strategies that survive the iterated deletion of strictly dominated strategies is necessarily no larger than the full set of strategy profiles If the game is dominance solvable, i.e. a single profile survives IDSDS then the solution must be a Nash equilibrium, so a dominant strategy equilibrium is a NE (e.g. Prisoner’s dilemma) but the opposite is not necessarily true (e.g. coordination game has NE but is not dominance solvable). If multiple (possibly all) strategies survive IDSDS then there will still be a Nash equilibrium (possibly in mixed strategies, as we will show later) slides_414_7_9_2011_no_solution 28 N‐PLAYER GAMES: MOTIVATION Some basic insights from games with small number of players extend to larger numbers, albeit with more complexity Some important issues only make sense to analyze with more than 2 players, e.g. Adoption of a standard or convention Potential number of entrants in a market Games with large number of players also highlight two important effects Tipping: if a large enough number of players adopts one strategy, e.g. standard or convention, than value of others doing the same increases. Congestion: force for playing different strategies (e.g. if all take one road or enter into a specific market then there is an incentive to use alternative strategy). As Yogi Berra put it: “That restaurant, nobody goes there anymore, it is too crowded”! slides_414_7_9_2011_no_solution 29 N‐PLAYER SYMMETRIC GAMES Definition: A game is symmetric when: All players have the same strategy sets, Players receive the same payoff when they choose the same strategy, Iftwo players’ strategies are switched, then their payoffs switch as well. Basic result in symmetric games: if an asymmetric strategy profile is a NE then so is a profile where players swap strategies E.g. n=2, chicken if we find (swerve, tough) is a NE then so is (tough, swerve) n=3: If (s’,s’’,s’’’) is a NE then so are all the other permutations (s’,s’’’,s’’), (s’’,s’,s’’’), (s’’,s’’’,s’) , (s’’’,s’,s’’) , (s’’’,s’’,s’) slides_414_7_9_2011_no_solution 30 A congestion question: Should Sneetches get starts upon thars? Simplified setup There are n sneetches and n is odd. A sneetch can costlessly add or remove a star upon thars Payoff highest if in the minority (i.e. stars are good if scarce, i.e. if # sneetches with stars, m<n/2 and otherwise no stars preferred) Exercise: How many sneetches will have stars painted on their belly? i.e. what are the Nash equilibria? slides_414_7_9_2011_no_solution 31 Players: two types (star and no star) Strategies For player s: remove star or do nothing For player ns: add star or do nothing Given symmetry we need only consider incentives to add or remove a star if m<n/2 Suppose for simplicity that n=7 and initially m=1 Is there an incentive for s to remove the star? No (1 to 0) Is there an incentive for one ns to add a star? Yes (0 to 1) Is there an incentive for another ns to add a star? Yes, since we would then still have m’<3.5=n/2 slides_414_7_9_2011_no_solution 32 When m’=3 what are the individual incentives? s does nothing (since if removed it would be in majority and get 0 instead of 1) ns has no incentive to add b/c his decision changes majority m’’=4>n/2 so gets 0 Two Nash equilibria for the symmetric sneetches game (n – 1)/2 sneetches have stars and (n + 1)/2 sneetches do not. Sneetches with stars have larger payoffs than those without stars. (n + 1)/2 sneetches have stars and (n – 1)/2 sneetches do not have stars. Sneetches without stars have larger payoffs than those with stars. slides_414_7_9_2011_no_solution 33 Congestion effect: at some point an increase in the number of sneetches with stars lowers the payoff of getting a star (similar point about no stars given the symmetry) in this case the congestion effect is discrete (since payoff discretely changes from 0 to 1 if in the minority) but more generally could be smoother, e.g. value of a star is inversely proportional to m. Other applications? slides_414_7_9_2011_no_solution 34 Network effects and Tipping points: Mac vs. Windows Definition: A product has network effects if its value to a consumer is greater when more consumers use it, e.g. email, a particular operating system Setup for examining n individual adoption decisions of Mac vs. Windows Payoff to adopting Mac: VM=100 + 10 m, where m = # Mac adopters VW= 10 x w, Payoff to adopting Windows: Underlying assumptions All n individuals adopt one or the other so m =n‐w and so VM=100 + 10 (nw) Mac is the better system in the absence of network effects Exercise: What are the potential candidates for Nash equilibria of this game? slides_414_7_9_2011_no_solution 35 Solution: … slides_414_7_9_2011_no_solution 36 A specific version of Mac vs. Windows, where n=20 All Mac and all Win still equilibria, as seen above Are there intermediate equilibria with m and w>0? No Incentives towards Mac when w=14<15 Incentive to move from win to Mac: VM(m’=7)=100+10x7 > 10x14 = VW(w=14) No incentive to move from Mac to win: VM(m=6)=100+10x6 > 10x15 = VW(w’=15) Also true for any w<14 Incentives towards Win when w=16>15 No incentive to move from win to Mac: VM(m’=5)=100+10x5 < 10x16 = VW(w=16) Incentive to move from Mac to win: VM(m=4)=100 + 10x4 < 10x17 = VW(w’=17) Tipping point: w=15 incentive to move in either direction Incentive to move from win to Mac: VM(m’=6)=100 + 10x6 > 10x15 = VW(w=15) Incentive to move from Mac to Win: VM(m=5)=100 + 10x5 < 10x16 = VW(w’=16) slides_414_7_9_2011_no_solution 37 Easier way to find the tipping point: plot payoffs for alternative options against the number of adopters of windows Find intersection: left payoffs to Mac higher; right those to Win are higher. Key role of expectations: if all exp...
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 Leftwing politics, Nash equilibria

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