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### Game Theory

Chapter 14

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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

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

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

- 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

- 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

- 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

- 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

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

- 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)

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

- 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

- 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

- 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

- 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

- 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

- 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

- 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

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

- 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

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 gamesstrategic 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

- 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

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

- 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

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

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

- 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

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

- 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

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

- 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

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