Game Theory DR. ÖZGÜR GÜRERK UNIVERSITY OF ERFURT WINTER TERM 2012/13. What can we model as a game?

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1 Game Theory 2. Strategic Games DR. ÖZGÜR GÜRERK UNIVERSITY OF ERFURT WINTER TERM 2012/13 What can we model as a game? Example: Firm behavior Players: Firms Actions: Prices that can be set by the firms Preferences: A reflection of the firms profits 2 Example: Candidates for a political office Players: Candidates Actions: Campaign expenditures Preferences: A reflection of candidates probabilities of winning

2 A note on modeling A model is an abstraction we use to understand our observations and experiences 3 What does understanding entail? Our perception of relationships between situations Isolating principles that apply to a range of problems so that we can fit into our thinking new situations that we encounter For example, a model of the movement of a tennis ball enhances our understanding because it fits well, independent of direction, type of ball The art of modeling Simplicity: Assumptions of a model should capture the essence of the situation, not irrelevant details Example: In the tennis ball model, we can ignore the dependence of gravitation force on the distance between the ball and the surface of earth 4 The models are neither absolutely right nor absolutely wrong Example: If you wish to find the shortest route between Erfurt and Jena you could assume that the earth is flat If you, however, ask the same question for San Fransisco and Mumbai, you should take into account that earth is spherical

3 Game-theoretic modeling Start with an idea related to some aspect of the interaction 5 Express the idea more precisely in a model Incorporate relevant features of the situation Put enough ingredients to obtain non-trivial insights but avoid irrelevant complications Analyze the model: Discover its implications i by logical l thinking The analysis may confirm our idea, or suggest it is wrong May be an assumption is inappropriate or some detail is missing We may conclude our idea is invalid or we try another model The interaction between ideas and models is bidirectional Strategic games A model of interacting players (DMs) 6 Each player has a set of possible actions Each player has preferences about the action profiles, i.e., the list of all players actions Example: A game with 2 players who each has 2 actions has 4 action profiles Assumption (for the moment): Ordinal preferences The model captures interaction between the players by allowing each player to be affected by the actions of all players

4 Further properties: p The payoff function 7 As in the theory of rational choice, payoff functions can be used to represent players preferences Suppose a player prefers the action profile a to profile b, and b to c These preferences may be specified by assigning 3 to a, 2 to b, and 1 to c or alternatively, by assigning 100 to a, 10 to b, and -1 to c A strategic game with ordinal preferences is defined by the preferences, not by payoffs that represent these preferences Time is absent from the model What is about time? Each player chooses her action for once and all 8 Players choose their actions simultaneously When choosing own action, no player is informed of the action chosen by any other player Nevertheless, an action may involve activities that may extend over time Example: If Apple s stock price falls below 400, buy 10 shares; otherwise do not buy any shares

5 Story: Example: The prisoners dilemma (PD) Two suspects in a major crime are questioned separately Enough evidence to convict each of them of a minor offense But not enough evidence to convict them of the major crime unless one of them acts as informer against the other (finks) 9 Consequences: If both stay quiet: One year prison for each If only one finks and the other stays quiet: The one who finks is freed, the other one gets 4 years If both fink: Each of them will spend 3 years in prison Modeling the prisoners dilemma Players: The two suspects Actions: Each player s set of actions is {quiet, fink} Profiles: (quiet, fink), (quiet, quiet), (fink, quiet), (fink, fink) Convention: The first (second) entry in a profile corresponds to the action chosen by player 1 (player 2) Preferences: How does suspect 1 order the action profiles? Assumption: She wants to spend as little time in prison as possible; this implies the ordering (from best to worst): (fink, quiet), (quiet, quiet), (fink, fink), (quiet, fink) Suspect 2 s ordering: (quiet, fink), (quiet, quiet), (fink, fink), (fink, quiet) 10

6 Choosing the payoff functions For player 1 we need a payoff function u 1 for which u 1 (fink, quiet)>u 1 (quiet, quiet)>u 1 (fink, fink)>u 1 (quiet, fink) 11 For example, we may specify u 1 as follows: u 1 (fink,quiet)=3, u 1 (quiet,quiet)=2, u 1 (fink,fink)=1, u 1 (quiet,fink)=0 For player 2, a payoff function u 2 must satisfy u 2 (quiet, fink)>u 2 (quiet, quiet)>u 2 (fink, fink)>u 2 (fink, quiet) For example, we may specify u 2 as follows: u 2 (quiet,fink)=3, u 2 (quiet,quiet)=2, u 2 (fink,fink)=1, u 2 (fink,quiet)=0 Representing the game 12 The rows: Actions of player 1 The columns: Actions of player 2 Suspect1 Suspect 2 quiet fink quiet 2, 2 0, 3 fink 3, 0 1, 1 Each box corresponds to an action profile Numbers in each box represent the players payoffs o Convention: The first (second) entry in a box corresponds to the payoff of player 1 (player 2)

7 Analyzing the prisoners dilemma A situation of cooperation and conflict Possible gains from cooperation (quiet, quiet) Each suspect, however, has an incentive to free-ride ride by finking whatever the other player does (recall both suspects sit in separate rooms) 13 Suspect 2 Suspect 2 quiet fink quiet fink Suspect 1 quiet 2, 2 0, 3 fink 3, 0 1, 1 Suspect 1 quiet 2, 2 0, 3 fink 3, 0 1, 1 Suspect 2 chooses fink whatever suspect 1 would choose What thinks Dilbert about PD? t / t h? 14

8 Why is PD interesting? In a variety of situations, players face similar incentives (cooperation and conflict) 15 An example from economics: a price-setting duopoly Each firm produces the same good Each firm wants to achieve the highest possible profit If each firm cares for the own profit than preferences can be represented by the obtained profits Firm 1 Firm 2 high low high 2M, 2M 0, 3M low 3M, 0M 1M, 1M Example for PD-like games: The arms race Suppose, two countries may build nuclear bombs or not 16 Assumption on preferences of each country (from best to worst) It has own bombs, opponent does not Neither country has any bombs Both countries have bombs Only the other country has bombs Country 2 This situation corresponds to PD No bombs = Quiet Build bombs = Fink Country 1 No bombs Build bombsb No bombs Build bombs 2, 2 0, 3 3, 0 1, 1

9 A brief history of game theory (GT) PD is first analyzed in early 1950s, the era of cold war Foundations of the field in The theory of games and economic behavior (1944) by J. von Neumann and O. Morgenstern John F. Nash, early 1950s, proposed an equilibrium i concept In the following years game theoretic models emerged in Economics, political science GT was used as a tool in evolutionary biology To study real behavior, experiments were conducted by economists and psychologists 17 Meanwhile, e, GT became e a very Nobel discipline dscp 1994 Prize to John Nash, John Harsanyi & Reinhard Selten for their pioneering analysis of equilibria in the theory of noncooperative games 2005 Prize to Robert Aumann & Thomas Schelling for having enhanced our understanding of conflict and cooperation through game-theory analysis 2009 Prize to Elinor Ostrom & Oliver Williamson for their analysis of economic governance, especially the commons (EO) and the boundaries of the firm (OW) Prize to Lloyd Shapley & Alvin Roth for the theory of stable allocations and the practice of market design

10 Other strategic games Example: Bach or Stravinsky? (Battle or Sexes: BoS) 19 Here, the players agree to cooperate; however, they disagree about the best outcome Other situations with similar incentives: P1 Bach P2 Stravinsky Bach 2, 1 0, 0 Stravinsky 0, 0 1, 2 Politics: Two officials i of a party deciding the stand to take Economics: Adapting a industry-wide standard for a new product Matching gpennies In this strictly competitive game, cooperation is not possible Two players choose, simultaneously, whether to show the head or the tail of a coin If they show the same side P2 pays P1 one Euro, if not P1 pays P2 one Euro Assumption: Each person cares for own monetary payoff 20 P1 P2 Head Tail Head 1, -1-1, 1 Tail -1, 1 1, -1 In such a game, players interests are diametrically opposed Example: Penalty kick

11 Nash equilibrium: Assumptions What actions will be chosen in a strategic game? Each player chooses the best available action based on the beliefs she forms about other players actions Beliefs are derived from past experience playing the game Players, however, view each play of the game in isolation 21 Imagine the idealized situation: For each player, there is a population p of DMs who may take that player s role In each instance of the game, players are selected randomly, one from each population Example: Buyers and sellers Many of the pairings may be modeled as random Nash equilibrium This solution theory has two components: Each player chooses her action according to the model of rational choice given her beliefs about other players actions Every player s belief about the other players actions is correct 22 Definition: A Nash equilibrium is an action profile a* with the property that no player i can do better by choosing an action different from a i *, given that every er other player j adheres to a j *

12 Nash equilibrium of the PD game Examine all four possible action pairs 23 Is (fink, fink) a Nash equilibrium? First, suppose suspect 1 chooses fink, what does suspect 2 choose then? Suspect 1 Suspect 2 quiet fink quiet 2, 2 0, 3 fink 3, 0 1, 1 Now, suppose suspect 2 chooses fink, what does suspect 1 choose then? Suspect 1 Suspect 2 quiet fink quiet 2, 2 0,3 fink 3, 0 1, 1 Nash equilibrium of the PD game For both players, fink is the best action given the other player chooses fink too Hence, the profile (fink, fink) is a Nash equilibrium i Is (quite, quite) a Nash equilibrium? 24 Suppose suspect 1 chooses quiet, what does suspect 2 choose then? Suspect 2 chooses fink, hence (quiet, quiet) can t be a Nash equilibrium Suspect 1 Suspect 2 quiet fink quiet 2, 2 0, 3 fink 3, 0 1, 1

13 Nash equilibrium of the PD game Is (fink, quite) a Nash equilibrium? 25 No, if suspect 1 chooses fink, suspect 2 would not choose quite, she would choose fink Suspect 1 Suspect 2 quiet fink quiet 2, 2 0, 3 fink 3, 0 1, 1 Hence, (fink, quite) no Nash equilibrium i Similar reasoning shows that (quite, fink) also no Nash equilibrium Nash equilibrium of the PD game Hence, (fink, fink) is the only Nash equilibrium of the PD game 26 The incentive to free-ride eliminates the mutually desirable cooperative outcome (quiet, quiet) Nash equilibrium of the PD-like examples we mentioned before: Price setting duopoly: (low, low) Arms race: (build bombs, build bombs)

14 Nash equilibrium: BoS (Bach, Bach): No player has an incentive to unilaterally deviate from this profile Nash equilibrium 27 (Bach, Stravinsky): Player 1 has an incentive to deviate (Stravinsky, Bach): Player 2 has an incentive to deviate P1 P2 Bach Stravinsky Bach 2, 1 0, 0 Stravinsky 0, 0 1, 2 (Stravinsky, Stravinsky): No player has an incentive to unilaterally deviate from this profile Nash equilibrium BoS has two Nash equilibria Nash equilibrium: Matching gpennies No Nash equilibrium 28 For action pairs (Head, Head) and (Tail, Tail) P2 is better of deviating P1 For action pairs (Head, Tail) and (Tail, Head) P1 is better of deviating P2 Head Tail Head 1, -1-1, 1 Tail -1, 1 1, -1 Later, we will return to this game