Game Theory Overheads

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

Hank should always choose 4 boxes Now what about Ken?

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

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

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

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

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

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

Members of cartels always have an incentive to cheat Why?

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

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."

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

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

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

The End