Game Theory

Learning Objectives

• Define game theory, and explain how it helps to better understand mutually interdependent management decisions

• Explain the essential dilemma faced by participants in the game called Prisoners’ Dilemma

• Explain the concept of a dominant strategy and its role in understanding how auctions can help improve the price for sellers, while still benefiting buyers

Overview

I. Introduction to Game TheoryII. Simultaneous-Move, One-Shot GamesIII. Infinitely Repeated GamesIV. Finitely Repeated GamesV. Multistage Games

Game Theory• Optimization has two shortcomings when applied to

actual business situations– Assumes factors such as reaction of competitors or

tastes and preferences of consumers remain constant.– Managers sometimes make decisions when other

parties have more information about market conditions.

• Game theory is concerned with “how individuals make decisions when they are aware that their actions affect each other and when each individual takes this into account.”

• Game Theory is a useful tool for managers

• In the analysis of games, the order in which players make decisions is important

• Simultaneous-move game- Each player makes decision without knowledge of other players decision

• Sequential-move game: player makes a move after observing other player’s move

• One shot game – underlying game is played only once

• Repeated game – underlying game is played more than once

• How managers use game theory:Betrand Duopoly game:2 gas stations – no location advantage.

Consumers view product as perfect substitutes and will purchase from station that sells at lower price.

First thing manager must do in the morning is to tell attendant to put up price without knowledge of rival’s price.

This is a simultaneous move game.If Manager of station A calls in price higher than B

will lose sales that day

Normal Form Game

• A Normal Form Game consists of:– Players.– Strategies or feasible actions.– Payoffs.

A Normal Form Game

Strategy A B Cabc

Player 2

Pla

yer

1 12,11 11,12 14,13

11,10 10,11 12,12

10,15 10,13 13,14

Simultaneous-move, One shot game

• Important to managers making decisions in an environment of interdependence. E.g. profits of firm A depends not only on firm’s A actions but on the actions of rival firm B as well.

Normal Form Game:Scenario Analysis

Strategy Left RightUp

Down

Player 2

Pla

yer

1 10,20 15,8

-10, 7 10,10

• What’s the optimal strategy?Complex question. Depends on the nature

game being played.

The game above is easy to characterize the optimal decision– a situation that involves a dominant strategy.

A strategy is dominant if it results in the highest payoff regardless of the action of the opponent

• For player 1, the dominant strategy is UP. Regardless of what player 2 chooses, if A chooses UP, she’ll earn more.

• Principle:Check to see if you have a dominant strategy. If

you have one, play it.

What should a player do in the absence of a dominant strategy (e.g. Player 2)?

Play a SECURE STRATEGY-- A strategy that guarantees the highest payoff

given the worst possible scenario.

Find the worse payoff that could arise for each action and choose the action that has the highest of the worse payoffs.

Secure strategy for player 2 is RIGHT. Guarantees a payment of 8 rather than 7 from LEFT

2 shortcomings:1.Very conservative strategy2.Does not take into account the optimal

decision of your rival and thus may prevent you from earning a significantly higher payoff.Player 2 should actually choose LEFT, knowing that player 1 will play UP

Principle: Put yourself in your rival’s shoesIf you do not have a dominant strategy, look at

the game from your rival’s perspective. If your rival has a dominant strategy, anticipate that she will play it.

Putting Yourself in your Rival’s Shoes

• What should player 2 do?– 2 has no dominant strategy!– But 2 should reason that 1 will play “a”.– Therefore 2 should choose “C”.

Strategy A B Cabc

Player 2

Pla

yer

1 12,11 11,12 14,13

11,10 10,11 12,12

10,15 10,13 13,14

The Outcome

• This outcome is called a Nash equilibrium:– “a” is player 1’s best response to “C”.– “C” is player 2’s best response to “a”.

Strategy A B Cabc

Player 2

Pla

yer

1 12,11 11,12 14,13

11,10 10,11 12,12

10,15 10,13 13,14

Nash Equilibrium

• Given the strategies of other players, no player can improve her payoff by unilaterally changing her own strategy.

• Every player is doing the best she can given what other players are doing.

• In original example, Nash equilibrium is when A chooses UP and B chooses LEFT.

Application of One shot games

• Two managers want to maximize market share.

• Strategies are pricing decisions. (charge high or low prices)

• Simultaneous moves.• One-shot game. (firms meet once and only

once in the market)

The Market-Share Game in Normal Form

Strategy Low Price High priceLow Price 0, 0 50, -10High Price -10,50 10,10

Manager 2

Man

ager

1

Market Share game Equilibrium

• Each manager’s best decision is to charge a low price regardless of the other’s decision. Outcome of game is that both firms charge a low price and earn 0 profits

• Low prices for both managers is the Nash Equilibrium

• If firms collude to charge high prices, profits will be higher for both

• Classic case in Economics called dilemma because the Nash equilibrium outcome is inferior (from the firms viewpoint) to the situation where they both “agree” to charge high prices

Even if firms meet secretly to collude, is there an incentive to “cheat” on the agreement?

To advertise or Not?• Your firm competes against another firm for

customers• You and your rivals know your product will be

obsolete at the end of the year (one shot game) and must simultaneously determine whether or not to advertise.

• In your industry, advertising does not increase industry demand but induces consumers to switch among the products of the different firms

An Advertising Game

Strategy Advertise No AdAdvertise 4,4 20,1

No Ad 1,20 10,10

Manager 2

Man

ager

1

To advertise or Not?• Dominant strategy of each firm is to advertise.

unique Nash equilibrium. • Collusion will not work because this is a one-

shot game and if there’s agreement not to advertise, each firm will have an incentve to cheat.

Key Insight:

• Game theory can be used to analyze situations where “payoffs” are non monetary!

• We will, without loss of generality, focus on environments where businesses want to maximize profits.– Hence, payoffs are measured in monetary

units.

Examples of Coordination Games

• Industry standards– size of floppy disks.– size of CDs.

• National standards– electric current.– traffic laws.

• Coordination Decisions:Firms don’t have competing objectives but

coordinating their decisions will lead to higher profits

e.g. Producing appliances that require either 90-volt or 120-volt outlets

A Coordination Game in Normal Form

Strategy 120-volt 90-volt120-volt 100,100 0,090-volt 0,0 100,100

Firm B

Fir

m A

Coordination Game: 2 Nash Equilibria

• What would you do if you manage Firm A?If you do not know what firm B is going to do,

you’ll have to guess what B will do. Effectively, both you and firm B will do better by

coordinating your actions.2 Nash equilibria. If the firms can ‘talk’ to each

other, they can agree on what to produce.Notice, there’s no incentive to cheat hereThis is a game of coordination rather than game

of conflicting interest

Simultaneous-Move Bargaining• Management and a union are negotiating a wage

increase.• Strategies are wage offers & wage demands.• Players have one chance to reach an agreement and

offer is made simultaneously.• Parties are bargaining over how much of $100 in

surplus must go to the union

• Assume the surplus can be split only into $50 increments

• One shot to reach agreement• Parties simultaneously write the amount they desire

on a piece of paper.• If the sum of the amounts does not exceed $100,

players get the specified amount• If sum exceeds $100, stalemate, costing each player $1

The Bargaining Game in Normal Form

Strategy 0 50 1000 0,0 0,50 0,10050 50,0 50,50 -1,-1

100 100,0 -1,-1 -1,-1

Union

Man

agem

ent

Simultaneous-Move Bargaining• 3 Nash equilibria outcomes.• Multiplicity of equilbria leads to inefficiency if parties

fail to “co-odinate” on an equilibrium• 6 of 9 outcomes are inefficient because they don’t sum

up to 100• Clearly, in this game management must ask for 50 if

they

Key Insights:

• Not all games are games of conflict. • Communication can help solve coordination

problems.• Sequential moves can help solve

coordination problems.

Infinitely Repeated Games• Game played over and over again. Players

receive payoff during each repetition of game• Firms compete week after week, year after

year game is repeated over time• To evaluate profits earned during this game,

consider the PV of all payoffs.• If payoffs are the same in each period, then

for an infinitely played game• PV = (1+i)/i * constant profit

An Advertising Game

• Two firms (Kellogg’s & General Mills) managers want to maximize profits.

• Strategies consist of pricing actions.• Simultaneous moves.

– Repeated interaction.

Equilibrium to the One-Shot Pricing Game

Strategy Low HighLow 0,0 50, -40High -40,50 10,10

General Mills

Kel

logg

’s

• When firms repeatedly face this type of matrix, they use “trigger strategy”

• Trigger Strategy – is a strategy that is contingent on the past plays of players in a game

• A player who adopts a trigger strategy continues to choose the same action until some other player takes an action that “triggers” a different action by the first player

Can collusion work if firms play the game each year, forever?

• Consider the following “trigger strategy” by each firm: – “We will each charge the high price, provided

neither of us has ever “cheated” in the past. If one of us cheats and charges a low price, the other player will “punish” the deviator by charging low price in ever period thereafter”

• In effect, each firm agrees to “cooperate” so long as the rival hasn’t “cheated” in the past. “Cheating” triggers punishment in all future periods.

Kellogg’s profits?

Cooperate = 10 +10/(1+i) + 10/(1+i)2 + 10/(1+i)3 + …

= 10 + 10/i Value of a perpetuity of $12 paid at the end of every year

Cheat = 50+0 +0 +0 +0

There’s no incentive to cheat if the PV from cheating is less than the PV from not cheating

Kellogg’s Gain to Cheating:• Cheat - Cooperate = 50 - (10 + 10/i) = 40 - 10/i

– Suppose i = .05• Cheat - Cooperate = 40- 10/.05 = 40 - 200 = -160• It doesn’t pay to deviate.

• As long as i is less than 25%, it pays not cheat.

– Collusion is a Nash equilibrium in the infinitely repeated game!

Benefits & Costs of Cheating• Cheat - Cooperate = 40 - 10/i

– 40 = Immediate Benefit (50 - 10 today) – 10/i = PV of Future Cost (10 - 0 forever after)

• If Immediate Benefit - PV of Future Cost > 0– Pays to “cheat”.

• If Immediate Benefit - PV of Future Cost 0– Doesn’t pay to “cheat”.

Application of Infinitely repeated games (product quality)

Strategy Low Quality HighDon't Buy 0,0 0, -10

Buy -10,10 1,1

Firm

Con

sum

ers

• If one shot game, Nash equilibrium = low quality product and don’t buy

• If infinitely repeated and consumers tell firm: “I’ll buy your product and will continue to buy if it is of good quality. But if it turns out to be shoddy, I’ll tell my friends not to buy anything from you again”.

• Given this strategy of consumers, what should the firm do?

• If the interest rate is not too high, the best alternative is to sell a high product quality

• If firm cheats and sells shoddy product, it will earn 10 now but 0 forever thereafter.

• It will not pay for the firm to cheat if the interest rate is low.

FINITE REPEATED GAMESGames that eventually end1.Games in which players do not know when the

game will end2.Games in which players know when it will end.

• Suppose two duopolists repeatedly play the pricing game until their product become obsolete. Suppose the firms don’t know when the game will end but there’s a probability p that the game will end after every given play

• Probability the game will be played tomorrow if played today is (1-p). If the game is played tomorrow, the probability it will be played the next day is (1-p)2 etc.

Pricing Game that is infinitely repeated

Strategy Low HighLow 0,0 50, -40High -40,50 10,10

General Mills

Kel

logg

’s

• Suppose firms adopt trigger strategies, whereby each agrees to charge a high price but if a firm deviates and charges a low price, the other firm will punish it by charging low price until the game ends.

• Assume interest rate is zero• Does Kellogg’s have an incentive to cheat?

Kellogg’s profits?

Cooperate = 10 +10/(1-p) + 10/(1-p)2 + 10/(1-p)3 + …

= 10/p

Cheat = 50+0 +0 +0 +0

There’s no incentive to cheat if the profit from cheating is less than the profit from not cheating. If there is a 10% that the government will ban the sale of the item, then profit from not cheating is 100 It pays not to cheat

Key Insight• Collusion can be sustained as a Nash

equilibrium when there is no certain “end” to a game.

• Doing so requires:– Ability to monitor actions of rivals.– Ability (and reputation for) punishing defectors.– Low interest rate.– High probability of future interaction.

End of Period Problem

• When players know precisely when a repeated game will end, end-of-period problem arises

• In the final period, there’s no tomorrow and there’s no way to punish a player for doing something wrong in the last period.

• Consequently, players will behave as if it was a one shot game

Resignations, Quits & Snake Oil salesmen• Workers work hard if threatened with being

fired if benefits of shirking are less than cost of being fired

• When worker announces that she wants to quit, say tomorrow, the cost of shirking is low so threat of firing has no effect

• What can managers do to overcome problem?1.Fire the worker as soon as she announces plan

to quit? ProblemsSnake Oil Salesmen move about so no

punishments

Factors affecting collusion in pricing games

• Number of firms: Collusion is easier when there are few firms rather than many.

• Firm Size: Economies of scale exists in monitoring. Easier for large firms to monitor small ones than other way round

• History of the Market: Explicit meeting to collude or tacit collusion?

• Punishment Mechanism: How do we punish our rivals when they cheat?

Real World Examples of Collusion• Garbage Collection Industry• OPEC• NASDAQ• Airlines

Multistage Games

• Timing is important• Players make sequential rather than

simultaneous decisions

A

UP

DOWN

B

B

UP

DOWN

UP

DOWN

The Game in Extensive Form

10, 15

5, 5

0, 0

6, 20

• Player A must make a decision before player B. A cannot make actions conditioned on what B does

• Player B’s action is dependent on what A does

• Suppose player B’s strategy is : Choose down if player A chooses up, and down if player A chooses down”

• Given this, best choice for A is down and earn 6

• If A chooses down, does B have an incentive to change her strategy? No these two strategies lead to a Nash equilibrium

• But this is not a subgame perfect equilibrium because there’s a superior Nash equilibrium

• Suppose B’s strategy is choose up if A chooses up and choose down if A chooses down, then A will choose up, earning 10, a better position than choosing down

• In the first equilibrium, A chose down because of B’s threats. But is the threat really credible?

Find the Subgame Perfect Nash Equilibrium Outcomes

• Outcomes where no player has an incentive to change its strategy, given the strategy of the rival, and

• The outcomes are based on “credible actions;” that is, they are not the result of “empty threats” by the rival.

Pricing to Prevent Entry: An Application of Game Theory

• Two firms: an incumbent and potential entrant.• Potential entrant’s strategies:

– Enter.– Stay Out.

• Incumbent’s strategies: – {if enter, play hard}.– {if enter, play soft}.– {if stay out, play hard}.– {if stay out, play soft}.

• Move Sequence: – Entrant moves first. Incumbent observes entrant’s action and

selects an action.

The Pricing to Prevent Entry Game in Extensive Form

Entrant

Out

Enter

Incumbent

Hard

Soft

-1, 1

5, 5

0, 10

Identify Nash and Subgame Perfect Equilibria

Entrant

Out

Enter

Incumbent

Hard

Soft

-1, 1

5, 5

0, 10

Two Nash Equilibria

Entrant

Out

Enter

Incumbent

Hard

Soft

-1, 1

5, 5

0, 10

Nash Equilibria Strategies {player 1; player 2}:{enter; If enter, play soft}{stay out; If enter, play hard}

One Subgame Perfect Equilibrium

Entrant

Out

Enter

Incumbent

Hard

Soft

-1, 1

5, 5

0, 10

Subgame Perfect Equilibrium Strategy:{enter; If enter, play soft}

Insights

• Establishing a reputation for being unkind to entrants can enhance long-term profits.

• It is costly to do so in the short-term, so much so that it isn’t optimal to do so in a one-shot game.

Games of ParticularRelevance in Economics

• Beach Kiosk Game– Two-Person, Zero-Sum, Non-cooperative– Example: two companies provide snacks and

sunscreen on a beach.• Beachgoers spread themselves out evenly along the

beach.• Both companies ultimately locate at the midpoint of

the beach, otherwise the other company has an advantage (closer to more beachgoers)

• Real life example: location of gas stations

Games of ParticularRelevance in Economics

• Repeated Game: game is played repeatedly over a period of time.

• In a repeated game, equilibria that are not stable may become stable due to the threat of retaliation.

Games of ParticularRelevance in Economics

• Repeated Game: game is played many times, and equilibria that are not stable may become stable due to the threat of retaliation.

• Assume (High, High) equilibrium reached and both firms start off charging the high price.

• In the next period, if one firm cheats (charges low price), it receives 600 in that period.

• Other firm will change to low prices in the next period to “retaliate” and both will end up at (Low, Low) equilibrium.

• Thus, incentive exists not to “cheat” in a repeated game and (High, High) is a viable equilibrium, though it is not in a single-period game.

• If number of periods are fixed, both firms will have incentive to cheat (charge low price) in the last period due to lack of threat of retaliation, which will then allow them to cheat in all periods.

Games of ParticularRelevance in Economics

• Consider the following payoff matrix in which firms choose their capacity, either high or low.

• Suppose firm C has the ability to move first.– C would choose Low, then D would choose High.

Game Theory and Auctions• Non-cooperative, non-zero-sum game• Seller wants to sell at highest price, buyer wants to

buy at lowest price.• Dutch Auction

– All product sold at the highest price that clears the market– Each buyer describes the quantity demanded and price to

pay– Starting at highest price, sum quantity demanded up to

the quantity available. The associated price for the last quantity added is the price for all products.

• In an auction with a time limit, every player has a dominant strategy to bid as late as possible.

Strategy and Game Theory• In Prisoners’ Dilemma, players have a dominant

strategy that leads to suboptimal results.• Commitment, explicit or implicit, can be used to

achieve preferred outcomes.• Commitment must be credible to have effect.• To make a commitment credible:

– Burn bridges behind you.– Establish and use a reputation.– Write contracts.

• Incentives also can be used to change the game to achieve preferred outcomes.

Strategy and Game Theory• Fundamental aspects of game theory

– Players are interdependent– Uncertainty: other players’ actions are not entirely

predictable• PARTS: paradigm for studying a situation, predicting

players’ actions, making strategic decisions– Players: Who are players and what are their goals?– Added Value: What do the different players contribute to

the pie?– Rules: What is the form of competition? Time structure of

the game?– Tactics: What options are open to the players?

Commitments? Incentives?– Scope: What are the boundaries of the game?