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Game Theory

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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

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Overview I.Introduction to Game Theory II. Simultaneous-Move, One-Shot Games III. Infinitely Repeated Games IV. Finitely Repeated Games V. Multistage Games

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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

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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

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One shot game – underlying game is played only once Repeated game – underlying game is played more than once

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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

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

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A Normal Form Game Player 2 Player 1 12,11 11,1214,13 11,1010,1112,12 10,1510,1313,14

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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.

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Normal Form Game: Scenario Analysis Player 2 Player 1 10,20 15,8 -10, 710,10

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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

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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.

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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.

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Secure strategy for player 2 is RIGHT. Guarantees a payment of 8 rather than 7 from LEFT 2 shortcomings: 1.Very conservative strategy 2.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

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Principle: Put yourself in your rival’s shoes If 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.

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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”. Player 2 Player 1 12,11 11,1214,13 11,1010,1112,12 10,1510,1313,14

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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”. Player 2 Player 1 12,11 11,1214,13 11,1010,1112,12 10,1510,1313,14

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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.

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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)

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The Market-Share Game in Normal Form Manager 2 Manager 1

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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

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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?

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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

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An Advertising Game Manager 2 Manager 1

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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.

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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.

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Examples of Coordination Games Industry standards – size of floppy disks. – size of CDs. National standards – electric current. – traffic laws.

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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

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A Coordination Game in Normal Form Firm B Firm A

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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 here This is a game of coordination rather than game of conflicting interest

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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

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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

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The Bargaining Game in Normal Form Union Management

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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

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Key Insights: Not all games are games of conflict. Communication can help solve coordination problems. Sequential moves can help solve coordination problems.

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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

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An Advertising Game Two firms (Kellogg’s & General Mills) managers want to maximize profits. Strategies consist of pricing actions. Simultaneous moves. – Repeated interaction.

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Equilibrium to the One-Shot Pricing Game General Mills Kellogg’s

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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

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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.

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Kellogg’s profits? Cooperate = /(1+i) + 10/(1+i) /(1+i) 3 + … = /i Value of a perpetuity of $12 paid at the end of every year Cheat = There’s no incentive to cheat if the PV from cheating is less than the PV from not cheating

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Kellogg’s Gain to Cheating: Cheat - Cooperate = 50 - ( /i) = /i – Suppose i =.05 Cheat - Cooperate = /.05 = = -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!

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Benefits & Costs of Cheating Cheat - Cooperate = /i – 40 = Immediate Benefit ( today) – 10/i = PV of Future Cost ( 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”.

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Application of Infinitely repeated games (product quality) Firm Consumers

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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

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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.

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FINITE REPEATED GAMES Games that eventually end 1.Games in which players do not know when the game will end 2.Games in which players know when it will end.

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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.

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Pricing Game that is infinitely repeated General Mills Kellogg’s

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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?

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Kellogg’s profits? Cooperate = /(1-p) + 10/(1-p) /(1-p) 3 + … = 10/p Cheat = 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

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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.

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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

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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? Problems Snake Oil Salesmen move about so no punishments

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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?

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Real World Examples of Collusion Garbage Collection Industry OPEC NASDAQ Airlines

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