Class 10 Supplement – Repeated Games and Mixed Strategies
Infinitely Repeated Games (Overcoming the Prisoner’s Dilemma)
An infinitely repeated game is a game that is played over and over again forever and in which players receive payoffs during each play of the game.
Overt (Explicit) Versus Tacit (Implicit) Collusion
Even if overt collusion is illegal, the possibility of enforcement in an infinitely repeated game may be sufficient to support tacit collusion.
Supporting Tacit Collusion with Trigger Strategies
A trigger strategy is a strategy that is contingent on the past plays of players in a game. A player using 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.
Review of Present Value
The present value of future profit flows is
If n goes to infinity and pt is constant, then
Trigger Strategies
· Grim Strategy: The firm: (1) cooperates in the first period, (2) cooperates in every succeeding period, provided the other firm does likewise, and (3) deviates forever after, if the other firm ever deviates. A grim strategy will successfully enforce collusion if
· Tit-for-Tat Pricing Strategy: The firm: (1) cooperates in the first period and (2), in each succeeding period, echoes (i.e., imitates) the competitor's previous action.
· And many more.
Trigger Pricing Strategies
· Grim Pricing Strategy: The firm: (1) sets a high price in the first period, (2) sets a high price in every succeeding period, provided the other firm does likewise, and (3) sets low prices forever after, if the other firm ever charges a low price.
· Tit-for-Tat Pricing Strategy: The firm: (1) sets a high price in the first period and (2), in each succeeding period, echoes (i.e., imitates) the competitor's previous price.
Applications
- An Infinitely Repeated Game: Pricing Decisions (Attached)
- An Infinitely Repeated Game: Advertising Decisions (Attached)
Finitely Repeated Games
There are two types of finitely repeated games that we will cover. The first is repeated games with a known final period that is referred to as the "end-of-period problem." The second is repeated games with an uncertain final period.
Applications
- Repeated Games with a Known Final Period: Another Application to Pricing Decisions (Attached)
- Repeated Games with an Uncertain Final Period: Yet Another Application to Pricing Decisions (Attached)
Sequential Games (Extensive Form Games)
Applications
· Simultaneous versus Sequential Games: The Case of Entry (Attached)
· Does It Matter Whom Moves First? (Attached)
Mixed Strategy Games
Application
· McDonalds's versus Burger King (Attached)
An Infinitely Repeated Game: Pricing Decisions
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Chevrolet (GM) |
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Low Price |
High Price |
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Ford |
Low Price |
0, 0 |
50, -10 |
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High Price |
-10, 50 |
10, 10 |
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Can Ford and Chevrolet tacitly (implicitly) collude on charging a high price? Suppose both Ford and Chevrolet adopt a grim strategy for enforcing cooperation. For example, Ford will: (1) choose a high price in the first period, (2) choose a high price in every succeeding period, provided Chevrolet does likewise, and (3) choose a low price forever if Chevrolet ever charges a low price. Assume the interest rate is 10%, or i = 0.10.
Comparing the Present Value of Cooperation versus Cheating
Calculate the present value of the cooperative solution.
Calculate the present value of the cheating solution given the other firm adopts a grim strategy.
Comparing the Difference Between Cheating and Cooperation with the Present Value of the Difference Between Cooperation and Noncooperation
A grim strategy will successfully enforce cooperation if 
.
Calculating the Maximum Interest Rate That Results in Cooperation.
Why Has Pricing Strategy in the Automobile Industry Changed in the Past Few Years?
An Infinitely Repeated Game: Advertising Decisions
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General Mills |
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Advertise |
Don’t Advertise |
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Kellogg |
Advertise |
1, 1 |
1.5, 0.5 |
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Don’t Advertise |
0.5, 1.5 |
1.25, 1.25 |
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Can Kellogg and General Mills tacitly (implicitly) collude on not advertising? Suppose both Kellogg and General Mills adopt a grim strategy for enforcing cooperation. For example, Kellogg will (1) not advertise in the first period, (2) not advertise in every succeeding period, provided General Mills does likewise, and (3) advertise forever if General Mills ever advertises. Assume the interest rate is 10%, or i = 0.10.
Comparing the Present Value of Cooperation versus Cheating
Calculate the present value of the cooperative solution.
Calculate the present value of the cheating solution given the other firm adopts a grim strategy.
Comparing the Difference Between Cheating and Cooperation with the Present Value of the Difference Between Cooperation and Noncooperation
A grim strategy will successfully enforce cooperation if 
.
Calculating the Maximum Interest Rate That Results in Cooperation.
Why do Kellogg and General Mills Both Do Heavy Advertising in Reality?
Repeated Games with a Known Final Period: Another Application to Pricing Decisions
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Chevrolet (GM) |
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Low Price |
High Price |
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Ford |
Low Price |
0, 0 |
50, -10 |
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High Price |
-10, 50 |
10, 10 |
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Suppose Ford and Chevrolet prefer to tacitly (implicitly) collude on charging a high price. In an infinitely repeated game, we found the grim strategy could enforce cooperation. But, what happens if the players know the game will end at some point in time?
Repeated Games with an Uncertain Final Period: Yet Another Application to Pricing Decisions
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Chevrolet (GM) |
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Low Price |
High Price |
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Ford |
Low Price |
0, 0 |
50, -10 |
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High Price |
-10, 50 |
10, 10 |
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Suppose Ford and Chevrolet prefer to tacitly (implicitly) collude on charging a high price. In a finitely repeated game with a known final period, we found the cooperative solution could not be maintained. But, what happens if the players are not sure when the game will end?
Comparing the Expected Value of Cooperation versus Cheating
Let q represent the probability that the game will not be played again. Thus, 1 - q is the probability that the game will be played again. For simplicity, assume the interest rate is 0.
Letting q = 15%, calculate the expected value of the cooperative solution.
Calculate the expected value of the cheating solution given the other firm adopts a grim strategy.
Calculating the Maximum Probability the Game Will End That Results in Cooperation.
Simultaneous versus Sequential Games: The Case of Entry
Suppose that we consider a monopolist who is facing a threat of entry by another firm. The entrant decides whether or not to come into the market, and the incumbent decides whether or not to cut its price in response.
If the entrant decides to stay out, it gets a payoff of 0 and the incumbent gets a payoff of 12. If the entrant decides to come in, then its payoff depends on whether the incumbent fights – by competing vigorously – or not. If the incumbent decides not to fight, we suppose that the entrant gets 4 and the incumbent gets 6. If the incumbent fights, the entrant gets -4 and the incumbent gets 4.
Entry Decision (Simultaneous Decisions)
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Intel |
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Don't Fight |
Fight |
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Babson
Microprocessor |
Stay Out |
0, 12 |
0, 12 |
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Enter |
4, 6 |
-4, 4 |
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Does It Matter Whom Moves First?
Entry Decision – Babson Moves First (Sequential Decisions)
Sub-game perfect equilibrium:
Entry Decision – Intel Moves First (Sequential Decisions)
Sub-game perfect equilibrium:
McDonald's versus Burger King
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Burger King |
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Low Price |
Status Quo |
Heavy Advertising |
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McDonald's |
Low Price Status Quo Heavy Advertising |
60, 35 |
65, 20 |
55, 45 |
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40, 40 |
60, 40 |
45, 55 |
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55, 50 |
60, 30 |
60, 40 |
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After eliminating Dominated Strategies, the game reduces to
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Burger King |
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Low Price |
Heavy Advertising |
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McDonald's |
Low Price |
60, 35 |
55, 45 |
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Heavy Advertising |
55, 50 |
60, 40 |
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McDonald's:
pBL = Probability Burger King chooses Low Price
pBH = Probability Burger King chooses Heavy Advertising
pBH = (1- pBL)
E[PML] = E[PMH]
[pBL´___] + [(1 - pBL)´___] = [pBL´___] + [(1 - pBL)´___]
Burger King:
pML = Probability McDonald's chooses Low Price
pMH = Probability McDonald's chooses Heavy Advertising
pMH = (1- pML)
E[PBL] = E[PBH]
[pML´___] + [(1 - pML)´___] = [pML´___] + [(1 - pML)´___]