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Game Theory Overheads

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A game is a formal representation of a situation in which a number of decision makers (players) interact in a setting of strategic interdependence Games By that, we mean that the welfare of each decision maker depends not only on her own actions, but also on the actions of the other players

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Structure of a Game Players or decision makers firms consumers poker or chess players

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Rules of the game Structure of a Game Who moves when What do players know and when What actions are available at various points

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For each possible set of actions by the players, what is the outcome of the game? Structure of a Game Outcomes

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Structure of a Game Payoffs What are the payoffs with each outcome? How do the player's rank the outcomes?

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Players or decision makers Structure of a Game Rules of the game Outcomes Payoffs

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Nash Equilibrium and Game Theory A set of strategies in a game is called a Nash equilibrium if, holding the strategies of all other firms constant, no firm can obtain a higher payoff by choosing a different strategy In a Nash equilibrium, no firm wants to change its strategy

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The Prisoner’s Dilemma Two criminals Al “Scarface” Capone Jack “Machine Gun” McGurn Major crime Murder of 7 people on St. Valentine’s Day

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Minimal evidence connecting them to massacre Plenty of evidence linking them to bootlegging

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Police interrogate them in separate rooms Police offer them each a deal Finger the other guy, and walk if he don’t talk You both stay mum, we lock the both of you up for a year for bootlegging You both talk, its 20 years in the slammer But if he talks and you give us the silent treatment, its 40 years at hard labor for you

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The Game in Matrix Form Scarface Stay Mum Machine Gun Al gets 20 Jack gets 20 Collaborate Stay Mum Collaborate Al walks Jack gets 40 Al gets 40 Jack walks Al gets 1 Jack gets 1

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Rules of the game Structure of a Game Who moves when -- at the same time What do players know and when -- nothing What actions are available at various points Collaborate Stay mum

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Outcomes Al gets 20 Jack gets 20 Scarface Stay Mum Machine Gun Collaborate Stay Mum Collaborate Al walks Jack gets 40 Al gets 40 Jack walks Al gets 1 Jack gets 1 Structure of a Game

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Payoffs What are the payoffs with each outcome? How do the player's rank the outcomes? Years in the slammer for the player Players prefer less years in the pen

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Best response for Al “Scarface” Capone What if Jack stays mum?

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Stay Mum Machine Gun Scarface Stay Mum Collaborate Al walks Jack gets 40 Al gets 1 Jack gets 1 Jack stays mum

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Al gets the least years by collaborating What if Jack collaborates?

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Scarface Stay Mum Collaborate Jack collaborates Machine Gun Collaborate Al gets 20 Jack gets 20 Al gets 40 Jack walks

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Al has found a dominant strategy for this game Al sells out and collaborates Al gets the least years by collaborating

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Best response for “Machine Gun” McGurn What if Al stays mum?

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Scarface Stay Mum Al stays mum Al gets 40 Jack walks Al gets 1 Jack gets 1 Machine Gun Stay Mum Collaborate

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Jack gets the least years by collaborating What if Al collaborates?

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Al collaborates Machine Gun Stay Mum Collaborate Scarface Collaborate Al gets 20 Jack gets 20 Al walks Jack gets 40

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Jack gets the least years by collaborating Jack has found a dominant strategy for this game Jack sells out and collaborates

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In fact Both Al and Jack sell out and get 20 years

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Dominant strategies A strategy that is best for a player, no matter what strategy is chosen by the competing player is called a dominant strategy A dominant strategy is a strategy that is best for a player regardless of the strategy of the other player

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If each player in a game has a dominant strategy, it is easy to find the equilibrium of the game If a player has a dominant strategy in a game, we can assume that the player will play that strategy It is simply the outcome that occurs when each player plays the dominant strategy

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Equilibrium in the prisoner’s dilemma Al and Jack get 20 years If both stayed mum, each would get 1 year

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The Turkey Pricing game Two stores Hy-Vee Fareway Rules of the game Game played 7 days before Thanksgiving Turkeys are the same quality

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Stores place a weekly advertising supplement in the local paper stating a price for turkeys Two possible actions for each store Low Price High Price The stores do not know the price that will be submitted by the other store

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Outcome Matrix Fareway High Price Hy-Vee F 1000 H 1000 Low Price High Price Low Price F 1500 H 500 F 500 H 1500 F 1200 H 1200

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Payoffs Each store gets the number in the table in dollars Store managers prefer more dollars to less

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Best response for Fareway What if Hy-Vee chooses a low price?

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Hy-Vee chooses low Hy-Vee Low Price Fareway High Price Low Price F 1000 H 1000 F 500 H 1500

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Fareway gets the most dollars with a low price What if Hy-Vee chooses a high price?

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Hy-Vee chooses high Fareway High Price Low Price Hy-Vee High Price F 1500 H 500 F 1200 H 1200

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Fareway gets the most money with a low price Fareway has a dominant strategy for this game Fareway chooses a low price

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Best response for Hy-Vee What if Fareway chooses a low price?

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Fareway chooses low Fareway Low Price Hy-Vee Low Price High Price F 1000 H 1000 F 1500 H 500

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Hy-Vee gets the most dollars with a low price What if Fareway chooses a high price?

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Fareway chooses high Hy-Vee Low Price High Price Fareway High Price F 500 H 1500 F 1200 H 1200

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Hy-Vee gets the most money with a low price Hy-Vee has a dominant strategy for this game Hy-Vee chooses a low price

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Equilibrium in the turkey pricing game Hy-Vee and Fareway choose low prices If both chose high prices, individual and total profits would be higher The outcome (high, high) is Pareto superior to the outcome (low, low)

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An outcome is said to be Pareto superior to another outcome if both both players are better off with the Pareto superior outcome

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Other examples of the prisoner’s dilemma Arms race Advertising tobacco eye-glasses Common resources oil field pasture

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Noncooperative oligopoly as a game Zucchini pricing game 2 players or firms -- Hank and Ken Actions: number of zucchini brought to market Firm doesn’t know other firm’s quantity Outcome is market price Payoff to each firm is its profit given price and costs

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Outcome Matrix Hank Ken H $36 K $36 4 boxes 3 boxes H $30 K $40 H $40 K $30 H $32 K $32 3 boxes 4 boxes

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Ken chooses 3 Hank Ken H $36 K $36 4 boxes 3 boxes H $40 K $30 3 boxes

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Ken chooses 4 Hank Ken 4 boxes 3 boxes H $30 K $40 H $32 K $32 4 boxes

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Hank should always choose 4 boxes Now what about Ken?

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Hank chooses 3 Hank Ken H $36 K $36 3 boxes H $30 K $40 4 boxes

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Hank chooses 4 Hank Ken 3 boxes 4 boxes H $40 K $30 4 boxes H $32 K $32

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Ken should always choose 4 boxes Both of them should always choose 4 boxes The equilibrium is (4,4)

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Cooperative Oligopoly Explicit collusion An association of firms that explicitly agrees to coordinate its activities is called a cartel A cartel that includes all the firms in an industry is, in effect, a monopoly

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Zucchini Example with Collusion Explicit collusion will lead to (3,3) Profits are ($36, $36) as compared to ($32, $32)

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But the cartel will collapse Why? If Hank produces 3, Ken will want to produce 4 And vice versa

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Members of cartels always have an incentive to cheat Why?

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Explicit collusion is illegal in many countries Firms must therefore consider the benefits of collusion versus the costs of getting caught OPEC was a very successful cartel for many years

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Tacit Collusion Any time firms cooperate without an explicit agreement, they are engaging in tacit collusion Typically, players adopt strategies along the following lines: “In general, I will set a high price” “If my rival sets a low price this time, I will punish him by setting a low price next time."

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This is called a tit for tat strategy The idea is that the rival will catch on and both players will set a high price over time

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Another type of tacit collusion occurs when there is an acknowledged price leader in the industry This leader firm acts first and the others follow

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When cheating on the cartel is likely It is difficult to observe other firms’ prices Market demand is unstable There are a lot of other firms in the industry

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The End

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