Game Theory Topic 2 Simultaneous Games “Loretta’s driving because I’m drinking and I’m drinking because she’s driving.” - The Lockhorns

Review Understanding the game Noting if the rules are flexible Anticipating our opponents’ reactions Thinking one step ahead Where does this lead us? We’ve defined the “game” but not the outcome Mike Shor

Equilibrium The likely outcome of a game when rational, strategic agents interact Each player is playing his or her best strategy given the strategy choices of all other players No player has incentive to change his or her action unilaterally Outline: Model interactions as games Identify the equilibria Decide when they are likely to occur Mike Shor

Cigarette Advertising on TV All US tobacco companies advertised heavily on TV Surgeon General issues official warning Cigarette smoking may be hazardous Cigarette companies fear lawsuits Government may recover healthcare costs Companies strike agreement Carry the warning label and cease TV advertising in exchange for immunity from federal lawsuits. 1964 1970 Mike Shor

Strategic Interaction Players: Reynolds and Philip Morris Strategies: Advertise or Not Advertise Payoffs: Companies’ Profits Environment: Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game? Mike Shor

Strategic Form of a Game PLAYERS Philip Morris No Ad Ad Reynolds 50 , 50 20 , 60 60 , 20 30 , 30 STRATEGIES PAYOFFS Mike Shor

What to Do? Philip Morris No Ad Ad Reynolds 50 , 50 20 , 60 60 , 20 30 , 30 If you are advising Reynolds, what strategy do you recommend? Mike Shor

Best Replies A strategy is a best reply to some opponents’ strategy if it does at least as well as any other strategy si is a best reply to s-i if for every si’ Not necessarily unique Mike Shor

Solving the Game Best reply for Reynolds: Philip Morris No Ad Ad 50 , 50 20 , 60 60 , 20 30 , 30 Best reply for Reynolds: If Philip Morris advertises: If Philip Morris does not advertise: Mike Shor

Dominance A strategy is dominant if it outperforms all other strategies no matter what opposing players do Games with dominant strategies are easy to solve No need for “what if …” thinking Mike Shor

Dominance si strictly dominates si’ if for every s-i (the payoff is strictly higher for every strategy of the other players) si weakly dominates si’ if for every s-i, and for some s-i Mike Shor

Dominance A strategy si is strictly dominant if it strictly dominates all other strategies for that player A strategy si is weakly dominant if it weakly dominates all other strategies for that player Mike Shor

Dominance Example 1 A strictly dominates B & A strictly dominates C Therefore A is strictly dominant X Y Z A 10 20 30 B 8 18 25 C 5 Mike Shor

Dominance Example 2 A strictly dominates B & A weakly dominates C’ Therefore A is weakly dominant X Y Z A 10 20 30 B 8 18 25 C’ Mike Shor

Dominance Example 3 A strictly dominates B & A does not dominate C’’ Therefore A is not dominant X Y Z A 10 20 30 B 8 18 25 C’’ Mike Shor

Dominance If you have a dominant strategy use it. (and no ability to agree on an alternate course of action) use it. If your opponent has a dominant strategy then expect her to play it. Mike Shor

Prisoner’s Dilemma Both players have a dominant strategy Optimal No Ad Ad 50 , 50 20 , 60 60 , 20 30 , 30 Equilibrium Both players have a dominant strategy The equilibrium results in lower payoffs for each player Mike Shor

Prisoner’s Dilemma Both players have a dominant strategy (s1,s1) u11 , u11 u12 , π21 u21 , u12 u22 , π22 Both players have a dominant strategy (s1,s1) u11 > u21 u12 > u22 The equilibrium results in lower payoffs for each player u22 > u11 The above two statements imply: u12 > u22 > u11 > u21 Mike Shor

Cigarette Advertising After the 1970 agreement: Cigarette advertising decreased by $63 million Industry Profits rose by $91 million Mike Shor

Prisoner’s Dilemma The dominant strategy will be played Mike Shor

Social Behavior in Pigs Baldwin and Meese (1979), “Social Behavior in Pigs Studied by Means of Operant Conditioning,” Animal Behavior Two small pigs: First pig gets 8 units of food, second gets 2 If simultaneous, each gets 5 Pushing the lever costs 1 One small, one big: If big pig is first, eats all of the food If small pig is first, it gets 6 units of food If simultaneous, big pig gets 7 Mike Shor

Prisoner’s Dilemma The dominant strategy will be played An equilibrium is NOT necessarily efficient Players can be forced to accept mutually bad outcomes Bad to be playing a prisoner’s dilemma, but good to make others play Mike Shor

How to Win a Bidding War by Bidding Less? The battle for Federated (1988) Parent of Bloomingdales Current share price ≈ $60 Expected post-takeover share price ≈ $60 Macy’s offers $70/share contingent on receiving 50% of the shares Do you tender your shares to Macy’s? Mike Shor

How to Win a Bidding War (continued) Robert Campeau bids $74 per share not contingent on amount acquired Campeau’s Mixed Scheme: If less than 50% tender their shares, each receives: $74 per share If X>50% tender, each receives: Mike Shor

The Federated Game To whom do you tender your shares? Majority of Others Macy’s Campeau You $70 $60 $74 $67+ To whom do you tender your shares? Mike Shor

How to Win a Bidding War Each player has a dominant strategy: Tender shares to Campeau Resulting Price: (½ x 74) + (½ x 60) = $67 BUT: Macy’s offered $70 ! Mike Shor

“ The biggest, looniest deal ever. ” Dominant Strategies “ The biggest, looniest deal ever. ” – Fortune Magazine, July 1988 on Campeau’s acquisition of Federated Stores Mike Shor

Prisoner’s Dilemma Examples Pricing by Firms High or low prices? Value menus and loyalty programs Divorce Hire attorneys or proceed amicably? Nuclear Weapons Build or don’t build weapons? State governments Inducements to attract business to a state Mike Shor

Dominated Strategies Two restaurants compete Can charge price of $30, $50, or $60 Customer base consists of tourists and natives 600 tourists pick randomly 400 natives select the lowest price Marginal costs are $10 Mike Shor

Tourists & Natives Example scenario: Restaurant 1: $50, Restaurant 2: $60 Restaurant 1 gets: 300 tourists + 400 natives = 700 customers x ($50-$10) = $28K Restaurant 2 gets: 300 tourists + 0 natives = 300 customers x ($60-$10) = $15K Mike Shor

Tourists & Natives R. 2 $30 $50 $60 R. 1 10 , 10 14 , 12 14 , 15 12 , 14 20 , 20 28 , 15 15 , 14 15 , 28 25 , 25 in thousands of dollars Mike Shor

Dominance A strategy si is strictly dominated if some strategy si’ strictly dominates it A strategy si is weakly dominated if some strategy si’ weakly dominates it Mike Shor

Iterated Deletion of Strictly Dominated Strategies Does any player have a (strictly) dominated strategy? Eliminate the strictly dominated strategy Reduce the size of the game Repeat: Iterate the above procedure Mike Shor

Iterated Deletion of Dominated Strategies $30 $50 $60 R. 1 10 , 10 14 , 12 14 , 15 12 , 14 20 , 20 28 , 15 15 , 14 15 , 28 25 , 25 Mike Shor

No Dominated Strategies Often there are no dominated strategies Some games may have multiple equilibria Equilibrium selection becomes an issue Method: For each player, find the best response to every strategy of the other player Mike Shor

Equilibrium An outcome in which every player is playing a best response to the strategies of all other players. An equilibrium is a strategy profile s such that si is a best reply to s-i for all i. Mike Shor

Equilibrium Illustration The Lockhorns Mike Shor

Games of Coordination Complements & technology adoption Two complementing firms Must use same technology, but each firm has a preferred technology Equilibrium does not offer a unique prediction Commit (or go first) to win! Firm 2 A B Firm 1 100 , 50 0 , 0 0 , 0 50 , 100 Mike Shor

Games of Assurance Joint research ventures Each firm may invest $50,000 into an R&D project Project succeeds only if both invest If successful, each nets $75,000 Firm 2 $50K $0 Firm 1 75 , 75 -50 , 0 0 , -50 0 , 0 Mike Shor

Games of Chicken Entry into small markets Firm 2 Stay Swerve Firm 1 -50 , -50 100 , 0 0 , 100 50 , 50 Mike Shor

The Right Game to Play Why do we “solve” games? To know which one to play! How do internal corporate changes impact the outcome of strategic interaction? Some games are better than others Mike Shor

Capacity Constraints Can decreasing others’ added value increase our profits? Can decreasing total industry value increase our profits? Mike Shor

Multiple Equilibria What is the predictive power of game theory when there are multiple equilibria? Sometimes nothing ? Refinements Focal points Efficiency Evolutionary stability Fairness Risk dominance Mike Shor

Summary Games have predictable outcomes Select the right game to play Notice dominant & dominated strategies Select the right game to play Looking ahead: Sequential Games: How do games unfold over time? Mike Shor