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Game Theory:

Game Theory Chapter 14

Introduction:

2 Introduction Game theory considers situations where agents (households or firms) make decisions as strategic reactions to other agents’ actions (live variables) Instead of as reactions to exogenous prices (dead variables) One of the most general problems in economies is outguessing a rival For example, a firm seeks to determine its rival’s most profitable counterstrategy to its own current policy Formulates an appropriate defensive measure For example, in 1996 Pepsi supplied its cola aboard Russia’s space station Mir Coca-Cola countered by offering its cola aboard shuttle Endeavour In this chapter, we see how theory of how agents interact (called game theory ) has extended classical approach By considering in greater detail interaction among firms in oligopoly markets

Introduction:

3 Introduction Game theory provides an avenue for economists to investigate and develop descriptions of strategic interaction of agents Strategic interdependence Each agent’s welfare depends not only on her own actions but also on actions of other agents ( players ) Best actions for her may depend on what she expects other agents to do Theory emphasizes study of rational decision-making based on assumption that agents attempt to maximize utility Alternatively, agents’ behavior could be expanded by considering a sociological, psychological, or biological perspective Recent progress in game theory has resulted in ability to view economic behavior as a special case of game theory In economics, this strategic interdependence among agents is called noncooperative game theory Binding agreements among agents are not assumed Cooperation may or may not occur among agents as a result of rational decisions In contrast to cooperative game theory , where binding agreements are assumed For example, interaction of two football teams playing a game is non-cooperative In contrast, two people forming a loving relationship to jointly increase their welfare is a cooperative game

Introduction:

4 Introduction Strategic interdependence of perfectly competitive firms or a monopoly firm is either minor or nonexistent Models of perfect competition and monopoly do not require incorporating game theory In contrast, strategic interdependence is a major characteristic of imperfect competition Game theory has become the foundation of models addressing imperfect-competition firm behavior Economic models based on game theory are abstractions from strategic interaction of agents Allows tractable interactions, yielding implications and conclusions that can then be used for understanding actual strategic interactions In this chapter, we first develop both strategic and extensive forms of game theory In discussing Prisoners’ Dilemma we see difficulties of obtaining a cooperative solution without some binding agreement However, we show a cooperative solution may result if game is played repeatedly Prisoners’ Dilemma games assume that all players move simultaneously

Introduction:

5 Introduction An alternative set of games are sequential games One player may know other players’ choices prior to making a decision Within set of sequential games are preemption games Being first to make a move may have certain advantages Sometimes a player’s first move is to threaten other players We investigate consequences of idle threats One game theory model explains why people will generally drive their automobiles right through a green light Another investigates Prisoner’s Dilemma game with incomplete information Discuss possible mixed strategies for players to follow As a final application of game theory, we discuss quid pro quo Games are not resolved in isolation

The Game :

6 The Game Interaction among players is foundation of game theory The game is a model representing strategic interdependence of agents in a particular situation Strategic interdependence implies that optimal actions of a player may depend on what he expects other players will do Players are decision makers in game With ability to choose actions within a set of possible actions they may undertake Players may be an individual or group of households, firms, government, animals, or environment as a whole Number of players is finite Games are characterized by number of players (for example, a two-player or n-player game)

The Game :

7 The Game A game-theory model is composed of Players Rules by which game is played Rules involve what, when, and how game is played What information each player knows before she moves (chooses some action) When a player moves relative to other players How players can move (their set of choices) Outcome Payoffs Some reward or consequence of playing game May be in form of a change in (marginal) utility, revenue, profit, or some nonmonetary change in satisfaction Assumed that payoffs can at least be ranked ordinally in terms of each player’s preferences

The Game :

8 The Game An example of a game is the children’s hand game: Rock, Paper, Scissors Rules for game Each player simultaneously makes the figure rock, paper, or scissors with one of their hands Outcome Rock dominates (crushes) scissors, scissors dominate (cut) paper, and paper dominates (covers) rock In a two-person game, player who makes dominating figure wins the game When both make same figure, it’s a draw and neither player wins Players each develop strategies for playing game Strategy (also called a decision rule ) is set of actions a player may take Specifies how a player will act in every possible distinguishable circumstance in which he may be placed For example, how a firm will react to a competitor’s possible price changes is firm’s strategy for this competitor’s action In general, a strategy is a player’s action plan In Rock, Paper, Scissors, strategy is the decision about when to form a rock, paper, or scissors with one’s hand

The Game :

9 The Game A player’s strategy is his complete contingent plan If it could be written down, any other agent could follow the plan and duplicate player’s actions Thus, a strategy is a player’s course of action involving a set of actions (moves) dependent on actions of other players For instance with the game of chess, player develops a specific set of actions for each possible move her opponent could make Actions implement a given strategy

The Game :

10 The Game Strategic form lists set of possible player strategies and associated payoffs Table 14.1 shows strategic form for Rock, Paper, Scissors Strategy pairs consist of combination of strategies from the two agents If player F chooses rock and player R selects scissors Strategy pair is (rock, scissors) with outcome that rock crushes scissors Player F then wins and player R loses Strategies and payoffs can be summarized in a game matrix (a payoff matrix ) Lists payoffs for each player given their strategies In strategic form, only strategies are listed

Table 14.1 Strategic Form for the Rock, Paper, Scissors Game :

11 Table 14.1 Strategic Form for the Rock, Paper, Scissors Game

The Game :

12 The Game Extensive form provides an extended description of a game Reveals outcomes and payoffs from each set of player strategies and possible actions each player can take in response to other player’s moves Game tree is used to represent extensive form of a game Illustrated in Figure 14.1 for Rock, Paper, Scissors Game is played from left to right Each node (point) represents a player’s decision Connected by branches that indicate available actions a player Extensive form of a game can be used to model everything in strategic form plus information about sequence of actions and what information each player has at each node Contains more detailed information May help eliminate some possible equilibrium outcomes

Figure 14.1 Game tree for Rock, Paper, Scissors:

13 Figure 14.1 Game tree for Rock, Paper, Scissors

The Game :

14 The Game For example, in Figure 14.1, two players F and R have the action choice of making a rock, paper, or scissors If players move sequentially with player F moving first, player R can observe player F’s action and always win If at initial decision node (also called a root ) player F chooses rock Player R—observing player F’s choice—will choose paper Yields terminal node with an associated payoff Player F loses and player R wins Sequential moves put player who moves first at a disadvantage Other player will always choose an action that results in a win As a result of this disadvantage, player R will not reveal his action unless player F also reveals her action When players thus simultaneously reveal their actions, neither player has any prior information on the actions of the other player In a game of simultaneous moves, game tree can be constructed with either players’ actions at root

Equilibrium :

15 Equilibrium Market equilibrium exists when there is no incentive for agents to change their behavior Yields an equilibrium price and quantity In game theory, a similar equilibrium may exist where players have no incentives to change their strategy One equilibrium is called dominant strategy One strategy is preferred to another no matter what other players do When all players have a dominant strategy, an equilibrium of dominant strategies exists that is determined without a player having to consider behavior of other players However, usually a player must consider other players’ strategies May then reduce his set of strategy choices based on rational behavior By assuming all players are rational and attempting to maximizing utility, a player determines a rationalizable strategy Generally, players who do not believe in rationalizable strategies will attempt to maximize utility independent of other players

Equilibrium :

16 Equilibrium A unique equilibrium or a set of equilibria may occur within a set of strategies Called a Nash equilibrium (after mathematician John Nash) Each player’s selected strategy is his or her preferred response to strategies actually played by all other players Strategies are in a state of balance An equivalent definition of a Nash equilibrium is where each player’s belief about other players’ preferred strategies coincides with actual choice other players make No incentive on part of any players to change their choices In a two-player game, a Nash equilibrium is a pair of player strategies where strategy of one player is best strategy when other player plays his or her best strategy Not all games have a Nash equilibrium and some games may have a number of Nash equilibria

Strategic Form :

17 Strategic Form Strategic form of a game is a condensed version of extensive form Actions with each player’s strategy are not reported in strategic form (how you play is not reported) Only possible strategies of each player with associated payoffs (win or lose) are listed Initially we assume that both players possess perfect knowledge Each player knows his own payoffs and strategies and other player’s payoffs and strategies Each player knows that other player knows this In strategic form, a player’s decision problem is choosing his strategy given strategies he believes other players will choose Players simultaneously choose their strategies, and payoff for each player is determined For example, firms interacting within a market could compete in advertising or jointly advertise in an effort to increase total demand for their products In most economic situations, agents can jointly or independently influence total payoff Indicates a possibility of cooperation or collusion Collusion is a joint strategy that improves position of all players

Strategic Form :

18 Strategic Form An example of a strategic interaction among players is the Battle-of-the-Sexes game Strategic form of this game is presented in Table 14.2 Payoff matrix composed of (wife’s payoff, husband’s payoff) Two players are a wife and husband deciding what to do on a Saturday night Two choices: going to opera or to the fights If they both go to the opera (fights) they each receive some positive utility Wife’s (husband’s) level of satisfaction is higher than husband’s (wife’s) If husband goes to fights while the wife goes to the opera They each enjoy their respective activity but not as much as if they went together to either event If husband went to opera and wife to the fights Both receive disutility

Table 14.2 Battle-of-the-Sexes Game :

19 Table 14.2 Battle-of-the-Sexes Game

Strategic Form :

20 Strategic Form As shown in Table 14.2, sum of payoffs is higher in two strategy pairs where they go together to same event Compared with each going to a different event A result of payoffs is possibility of multiple Nash equilibria Both going to opera is a Nash equilibrium Because if either one picks fights instead their utility is decreased For example, if husband picks fights, his utility is reduced from 2 to 1 If wife picks fights, her utility falls from 5 to -7 Both going to fights is a Nash equilibrium If either one instead picks opera, wife’s utility falls from 2 to 1 and husband’s from 5 to -1 In general, even if a Nash equilibrium exists, it may not be unique Problem of multiple Nash equilibria can be avoided when players can choose a strategy mix

Prisoners’ Dilemma :

21 Prisoners’ Dilemma In general, Prisoners’ Dilemma game is a situation where two prisoners are accused of a crime D.A. does not have sufficient evidence to convict them Unless at least one of them supplies some supporting testimony If one prisoner were to testify against the other, conviction would be a certainty D.A. offers each prisoner separately a deal If one confesses while his accomplice remains silent Talkative prisoner will receive only 1 year in prison Silent prisoner will be sent up for maximum of 10 years If neither confesses, both will be prosecuted on a lesser offense If both confess, in which case testimony of neither is essential to the prosecution Both will be convicted of the major offense and sent up for 5 years As shown in Table 14.3, payoff matrix is composed of (F’s payoff, R’s payoff)

Table 14.3 Prisoners’ Dilemma:

22 Table 14.3 Prisoners’ Dilemma

Prisoners’ Dilemma :

23 Prisoners’ Dilemma Unique Nash equilibrium to Prisoners’ Dilemma is where each prisoner confesses and each is sentenced to 5 years From Table 14.3, if prisoner R does not confess, prisoner F can increase her payoff by confessing (reduced jail time by 1 year) If prisoner R confesses, prisoner F will again confess and receive 5 fewer years Thus, for prisoner F confessing is always preferred to not confessing Confessing is dominant strategy for prisoner F Confessing is also dominant strategy for prisoner R Thus, Nash equilibrium is both confessing No other pair of strategies is in Nash equilibrium If prisoner F does not confess, she will receive 10 years, because prisoner R will believe that if prisoner F confesses and he does not confess then he will receive 10 years Thus, prisoner R will confess Illustrates situation, common in economics, where cooperation (not confessing) can improve welfare of all players

Prisoners’ Dilemma :

24 Prisoners’ Dilemma Although dominant strategy of both confessing is Nash equilibrium strategy It is not preferred outcome of players acting jointly Both prisoners would prefer that they jointly do not confess and each receive only 2 years Classic example of rational self-serving behavior not resulting in a social optimum If the two prisoners could find a way to agree on the joint strategy of not confessing and, of equal importance, a way to enforce this agreement Both would be better off than when they play the game independently However, it is still in the interest of each prisoner to secretly break agreement One who breaks the deal and confesses will only receive 1 year while the other will pay price of receiving an additional 8 years Example of a bilateral externality

Enforcement :

25 Enforcement In Prisoners’ Dilemma example, Nash equilibrium results in confession when joint optimal solution would be for both prisoners to not confess For this joint cooperation to result, some type of enforcement is required Otherwise, there is an incentive on part of at least one player to break agreement Table 14.3 highlights difference between what is best from an individual’s point of view and that of a collective Conflict endangers almost every form of cooperation Reward for mutual cooperation is higher than punishment for mutual defection But a one-sided defection yields a temptation greater than that reward Leaves exploited cooperator with a loser’s payoff that is even worse than punishment for mutual defection Rankings from temptation through reward and punishment imply that the best move is always to defect, irrespective of the opposing player’s move Leads to mutual defection unless some type of enforcement exists

Cooperation :

26 Cooperation In general, agents attempt to cooperate Agents defecting from cooperative agreements are usually not observed in societies Agents often instead cooperate, motivated by feelings of solidarity or altruism In business agreements, defection is relatively rare Cooperation among agents in an economy may be as essential as competition for economic efficiency and enhancing social welfare A solution consistent with cooperation may result if Prisoners’ Dilemma game is repeatedly played If one player chooses to defect in one round, then other player can choose to defect in next round In a repeated game, each player has opportunity to establish a reputation for cooperation and encourage other player to cooperate If a game is repeated an infinite number of times Cooperative strategy of not confessing may dominate single-game Nash equilibrium of confessing

Cooperation :

27 Cooperation Consider first a finite number, T, of repeated games (a finitely repeated game ) Last round, T, is same as playing game once Solution will be the same and both players will defect by confessing In round (T - 1), there is no reason to cooperate since in round T they both defected Thus, in round (T - 1) they both defect Defection will continue in every round unless there is some way to enforce cooperation on last round However, if game is repeated an infinite number of times (an infinitely repeated game ) Player does have a way of influencing other player’s behavior If one player refuses to cooperate this time, other player can refuse to cooperate next time

Cooperation :

28 Cooperation Robert Axelrod identifies optimal strategy for an infinitely repeated game as tit-for-tat (also called a trigger strategy) On first round player F cooperates and does not confess On every round after, if player R cooperated on previous round, F cooperates If R defected on previous round, F then defects Strategy does very well because it offers an immediate punishment for defection and has a forgiving strategy An application is the carrot-and-stick strategy that underlies most attempts at raising children

Cooperation :

29 Cooperation An alternative strategy is win-stay/lose-shift If a player wins with a chosen strategy, she keeps same strategy for next round If she loses, she changes to an alternative strategy Similar to tit-for-tat strategy in terms of preventing exploiters from invading a cooperative society Will provide incentives for any exploiter to cooperate Exploiters in a cooperative society are players who attempt to maximize their payoff given strategies of other players Does not matter to exploiters if their strategy results in cooperation or not Only interested in maximizing their payoff However, this win-stay/lose-shift strategy fares poorly among noncooperators Against persistent defectors a player employing win-stay/lose-shift strategy tries every second round to resume cooperation

Sequential Games :

30 Sequential Games In a sequential, or dynamic, game, one player knows other player’s choice before she has to make a choice Many economic games have this structure For example, a monopolist can determine consumer demand prior to producing an output, or a buyer knows sticker price on a new automobile before making an offer As an example of a sequential game, consider Battle-of-the-Sexes game in Table 14.2 Husband prefers going to fights and wife prefers opera However, they both prefer spending their leisure time together Results in two pure-strategy Nash equilibria (both going to the opera or both to the fights) if both players reveal their choices simultaneously Suppose husband chooses first and then wife Game tree outlining this sequence of choices is illustrated in Figure 14.2 Game tree is a description of game in extensive form Indicates dynamic structure of game, where some choices are made before others Once a choice is made, players are in a subgame consisting of strategies and payoffs available to them from then on

Figure 14.2 Game tree for Battle-of-the- Sexes:

31 Figure 14.2 Game tree for Battle-of-the- Sexes

Sequential Games :

32 Sequential Games If husband picks opera, the subgame is for the wife to choose If she picks opera also, husband ends with a payoff of 2 and wife with a payoff of 5 If husband picks fights, it is optimal for wife to also pick fights Resulting payoffs are 5 for husband and 2 for wife For husband (first player), 5 is greater than 2 So equilibrium for this sequential game is for couple to go to the fights One of Nash equilibria in strategic form of the game, Table 14.2 Both going to the fights is not only an overall equilibrium, but also an equilibrium in each of the subgames A Nash equilibrium with this property is known as a subgame perfect Nash equilibrium Unique equilibrium of both going to the fights is conditional on who makes first choice

Sequential Games :

33 Sequential Games If instead wife made first move, alternative Nash equilibrium, both going to the opera, would be unique solution of this sequential game Thus, this strategy pair of opera and fights is really a subset of a larger game involving the strategies of moving first or second Use a technique called backward induction to determine a subgame perfect Nash equilibrium, by working backward toward the root in a game tree Once game is understood through backward induction, players play it forward To apply backward induction, first determine optimal actions at last decision nodes that result in terminal nodes Then determine optimal actions at next-to-last decision nodes, assuming that optimal actions will follow at next decision nodes Continue backward process until root node is reached Backward induction implicitly assumes that a player’s strategy will consist of optimal actions at every node in game tree Called principle of sequential rationality At any point in game tree, player’s strategy should consist of optimal actions from that point on given other players’ strategies

Figure 14.3 Reduced game tree for Battle-of- the-Sexes:

34 Figure 14.3 Reduced game tree for Battle-of- the-Sexes

Preemption Games :

35 Preemption Games Battle-of-the-Sexes game illustrates advantage of moving first In many economic game-theory models, firms who act first have an advantage Called preemption games  strategic precommitments can affect future payoffs For example, a firm adopting a relatively large production capacity in a new market can saturate market and make it difficult for ensuing firms to enter Any economies of scale associated with this production can be achieved with this large capacity Firm moving first has potential of lower average production costs Ability to seize a market first depends on market’s contestability If market is contestable, potential entrant firms can practice hit-and-run entry Will mitigate any advantages of moving first Governments concerned with ability of firms to saturate a market and forestall entry of other firms have attempted to place restrictions on such behavior Example: President Reagan placed a 5-year tariff on motorcycles to rescue domestic motorcycle company Harley-Davidson

Preemption Games :

36 Preemption Games An example of a preemption game is provided in Table 14.4 Firms 1 and 2 are faced with choice of entering or not entering a market Market is not large enough for both to enter, so if they both enter they will each experience losses in payoff of 5 If neither firm enters, both payoffs are 0 The two pure-strategy Nash equilibria are for one firm to enter and the other not Whichever firm moves first and enters market will receive a positive payoff of 10 Other firm will not enter and receive a 0 payoff Strategy for firms is to be first to enter market If one of the firms is a foreign firm and has some advantages of being first to enter a domestic market Domestic government may attempt to restrict that entry to enable domestic firm to enter first Once domestic firm enters, foreign firm no longer has an incentive to enter

Table 14.4 Preemption Game :

37 Table 14.4 Preemption Game

Market Niches :

38 Market Niches Preemption games can also help us understand discount stores’ location strategies In United States, small towns generally only have sufficient populations to support one major discount store First discount firm to establish a store in town drives out any pre-existing local nondiscount competition and has a local monopoly As country gets saturated with these discount stores, opportunities to establish local monopolies decline Discount firms will attempt to fill a market niche instead For example, Target stores cater to uppermiddle-income households Once a discount store enters a local market, existing nondiscount stores will attempt to adjust their market in an effort to find a market niche For nondiscount stores, price competing with a discount store is generally not an optimal choice

Market Niches :

39 Market Niches As implied in Table 14.5, a chain of discount stores will generally, by economies to scale, have lower average costs than a single nondiscount store If nondiscount store attempts to compete by lowering its price, discount store will also lower its price Results in losses for nondiscount store while discount store still remains profitable Dominant strategy for nondiscount store is to maintain its high price Strategy for discount firm is then to enter and offer slightly lower prices than nondiscount store Nondiscount store can then either develop a market niche around discount store or eventually go out of business

Table 14.5 Discount Entry :

40 Table 14.5 Discount Entry

Market Niches :

41 Market Niches In general, producers will attempt to occupy every market niche to keep potential entrants from gaining access into a market Through research and development, a firm will endeavor to supply a complete range of a particular product to cover every niche Consider two firms entertaining entry into a market for a commodity, say, breakfast cereals with two niches, sweet cereals, J, and healthy cereals H Payoff matrix is provided in Table 14.6 If both firms move simultaneously, two Nash equilibria result With each firm picking a different market niche Whichever firm moves first will capture preferred market niche and receive higher payoff To be first, the firm must make a commitment Either by actually providing product first or by advertising in advance that it will supply product for preferred niche If there are large sunk costs associated with this commitment, then the other firm (say, firm 2) will realize firm 1 is in fact committed to preferred product niche J Firm 2 may accede and supply in niche H

Table 14.6 Market Niches :

42 Table 14.6 Market Niches

Threats:

43 Threats Firm 1 could attempt to just threaten firm 2 Instead of making a commitment to supply in preferred niche market J and incurring sunk costs For example, firm 1 could threaten firm 2 by stating it will produce in niche J regardless of what firm 2 does However, firm 2 has to believe the threat to acquiesce One way to make a threat credible is to make commitment in sunk cost Or, firm 1 could simply mislead firm 2 into believing it is making a commitment to niche J when in fact it is not Assumes asymmetric information Idle or empty threats will not succeed in inducing a player to select some action

Threats:

44 Threats Consider two competing firms advertising Payoff matrix in Table 14.7 represents returns from firms’ choices of either advertising or not Pure-strategy Nash equilibrium is for firm 1 to advertise and firm 2 not to advertise Firm 1’s advertising has a relatively large impact on returns for the two firms In terms of advertising, firm 1 is dominant firm in industry Despite Firm 1’s dominance, firm 2’s advertising does positively affect firm 1’s returns By possibly expanding total market in which products are being advertised

Table 14.7 Idle Threats :

45 Table 14.7 Idle Threats

Threats:

46 Threats In this case, advertising is not drawing sales from one firm to another But instead is making product known to more consumers Enlarges both firms’ markets Thus firm 1 would prefer that firm 2 also advertise However, added expense of advertising by firm 2 is not covered by its returns However, even considering dominance of firm 1, it cannot threaten to not advertise in order to induce firm 2 into advertising Because no matter which choice firm 2 makes, firm 1’s dominant strategy and its subgame perfect Nash equilibrium is to advertise Firm 2 will realize that if firm 1 is rational it will always advertise, so a threat of not advertising by firm 1 is not credible Subgame perfect Nash equilibrium results in a selection of a Nash equilibrium obtained by removing strategies involving idle threats It is very important to always be willing and able to carry out a threat

Child Rearing :

47 Child Rearing If one player derives satisfaction from penalizing the other, threats made by player will be more credible The more credible the threat, the more likely it will be acted upon An example is child rearing Through reward and punishment, a parent derives satisfaction of good behavior from a child Figure 14.4 shows a game tree representing interactions of a parent and child Child selects her behavior and parent chooses to reward or punish it Pure Nash equilibrium is a badly behaved child rewarded Subgame perfect Nash equilibrium is for parent to always reward

Figure 14.4 A game tree for child rearing:

48 Figure 14.4 A game tree for child rearing

Child Rearing :

49 Child Rearing If child believes parent will always reward any behavior, it will choose bad behavior In contrast, if child is under impression that parent will punish bad behavior even if it hurts parent Threat by parent will not be idle In Figure 14.4, parent will not reward bad behavior even considering parent’s payoff increases from 35 to 40 Subgame perfect Nash equilibria are now for parent to reward good behavior and punish bad Child will then realize bad behavior will result in punishment with an associated zero payoff Child will select good behavior over bad and increase her payoff from 0 to 15 In general, this example of parent/child interaction is a principal/agent model, where principal is the parent and agent is the child Principal is attempting to provide incentives, both positive and negative, to elicit correct behavior from agent In a repeated game, consistent behavior on the part of a principal can dominate inconsistent behavior For example, if a parent is consistent in following through with any threats Child will realize that probability of punishment for bad behavior is high and correct her bad behavior

Child Rearing :

50 Child Rearing Establishing a reputation of always being committed to any threats can lead to cooperation by other player In Prisoners’ Dilemma game, an example of consistent behavior is where a tit-for-tat strategy is consistently played Unless these incentives (threats) are taken seriously, agent will not select principal’s desirable actions For example, suppose a pro-business governor relaxes regulatory constraints on small businesses by not enforcing various environmental regulations Threat of enforcement exists, but it is an idle threat If a pro-environmental governor is later elected Threat will become credible and firms will likely comply with regulations