1 1 deep thought ba 445 lesson b.4 simultaneous price competition as the light changed from red to...

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Deep Thought

BA 445 Lesson B.4 Simultaneous Price Competition

As the light changed from red to green to yellow and back to red again, I sat there thinking about life. Was it nothing more than a bunch of honking and yelling? Sometimes it seemed that way. ~ Jack Handey.

(Translation: Today’s lesson teaches how to anticipate the actions of other players, taking account that they are trying to anticipate your actions.)

2 2BA 445 Lesson B.4 Simultaneous Price Competition

Readings

Readings

Baye “Bertrand Oligopoly” (see the index)Dixit Chapter 4


Overview

Overview

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Overview


Forming Beliefs about others’ strategies that are chosen simultaneously with yours depends on assumptions of each player’s rationality, and each player’s beliefs of the rationality of other players.

Dominate Strategies for a player give better payoffs compared with any other strategy, no matter what other players choose for their strategies. Any rational player should choose a dominate strategy.

Weakly Dominate Strategies give at least as good payoffs compared with any other strategy, no matter what other players choose, and better payoffs for at least one choice of the other players.

Dominated Strategies for a player give worse payoffs compared with any other strategy, no matter what other players choose for their strategies. No rational player should choose a dominated strategy.

Weakly Dominated Strategies give at least as bad payoffs compared with any other strategy, no matter what other players choose, and worse payoffs for at least one choice of the other players.

Rationalizable Strategies are a best response to some belief about what other players choose, where the other players are, themselves, believed to choose rationalizable strategies.


Example 1: Forming Beliefs




Overview

Forming Beliefs about others’ strategies that are chosen simultaneously with yours depends on assumptions of each player’s rationality, and each player’s beliefs of the rationality of other players.



Comment: Sequential move games with perfect information typically have a unique rollback solution to choose your optimal strategy and to predict how others would have responded to every possible move you might have made.

When your opponents’ strategies are chosen simultaneously with yours, rather than sequentially, choosing your optimal strategies and forming beliefs about your opponents’ strategies can be harder since your opponents are simultaneously forming beliefs about you.

There are various solutions offered by game theory depending on then extent of players’ rationality and of assumptions about the rationality of other players.

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Question: How should you play the following Guess 2/3 of the Average Game?

Rules:1. No talking or other communication between players.2. Players secretly write a real number between 0 and

100.3. The winner is the one closest to 2/3 of the average.

What number should you guess?


Example 1: Forming Beliefs about Current Strategies

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Answer: Your optimal guess is 2/3 of the average of your beliefs about the guesses of the other players.

Game theory provides steps to form those beliefs based on a logical process of thinking through the thinking of the other players. You will put yourself in the position of other players and think through the others’ thinking, which of course includes their putting themselves in your position and thinking what you are thinking.



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Step 1: Any guess higher than 66.67 (2/3rds of 100) is worse for any player than guessing 66.67 since higher numbers cannot possibly be 2/3rds of the average of any guesses. Those higher numbers should be eliminated by any rational player.

Step 2: Once those guesses are eliminated for every player, any guess higher than 44.45 (2/3rds of 66.67) is worse for any player than guessing 44.45 since higher numbers cannot possibly be 2/3rds of the average of any remaining guesses (0 to 66.67) . Those higher numbers should be eliminated.

Step 3: Once those guesses are eliminated for every player, any guess higher than 29.64 (2/3rds of 44.45) is worse for any player than guessing 29.64 since higher numbers cannot possibly be 2/3rds of the average of any remaining guesses (0 to 44.45) . Those higher numbers should be eliminated.



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That process can continue until any particular number above 0 is eliminated. So, guess 0.



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Comment: Suppose you doubt the assumption of the common knowledge of the rationality of all players. For example, suppose you assume other players are rational, and you assume other players assume other players are rational, but you make no assumptions about the assumptions other players make about the assumptions of other players. What number should you guess?

Because you are rational, eliminate any guess higher than 66.67. Because you assume other players are rational, eliminate any

guess higher than 44.45. Because you assume other players assume other players are

rational, eliminate any guess higher than 29.64. Without further assumptions, all guesses from 0 to 29.64 are

viable.




Example 2: Dominate Strategies



Overview

Dominate Strategies for a player give better payoffs compared with any other strategy, no matter what other players choose for their strategies. Any rational player should choose a dominate strategy.



Comment: Game tables or normal forms condense the information in a game tree or extensive form. Like the extensive form, the normal form specifies strategies for every player and the outcomes of the actions taken by all players. But unlike the extensive form, the normal form does not specify the order of the actions. Normal forms are the simplest way to model games where actions are simultaneous.


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Question: Sam’s Club and Costco control a large share of the US retail wholesale market. Both sell emergency food supplies in a weather-proof bucket that provides 275 delicious easy-to-prepare meals, including potato soup and corn chowder. The unit cost to both retailers is $75. The retailers compete on price: the low-price retailer gets all the market and they split the market if they have equal prices. Suppose they simultaneously consider prices $85 and $95, and suppose market demands at those prices are 100 and 80.

Determine optimal prices in this Price Competition Game.

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Answer: $85 is a dominate strategy for each player since it gives better payoffs for that player compared with $95, no matter whether the other player chooses $85 or $95.

$85 $95$85 500,500 1000,0$95 0,1000 800,800

Costco

Sam's

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$85 $95$85 500,500 1000,0$95 0,1000 800,800

Costco

Sam's

Confess Don't C.Confess 500,500 1000,0Don't C. 0,1000 800,800

Prisoner 2

Prisoner 1

Comment: The Reduced Price Competition Game is like the famous prisoners’ dilemma.

The prisoner's dilemma is a fundamental problem in game theory that demonstrates why two people might not cooperate even if it is in both their best interests to do so.

Two suspects are arrested. Each is told by the police they are best off if they confess, making confession a dominate strategy. But both prisoners’ confessing is worse for each than both not confessing.


Example 3: Weakly Dominate Strategies



Overview

Weakly Dominate Strategies give at least as good payoffs compared with any other strategy, no matter what other players choose, and better payoffs for at least one choice of the other players.



Comment 1: A weakly dominate strategy for a player gives at least as good payoffs for that player compared with any other strategy, no matter what other players choose for their strategies, and better payoffs for at least one choice of strategies for the other players. Any rational player has no reason not to choose a weakly dominate strategy. And a rational player should definitely choose it if there is any positive probability belief attached to those strategies for the other players that make the weakly dominate strategy give better payoffs. Thus, a rational player should definitely choose a weakly dominate strategy if there is any positive probability belief that the other players, through a slip of the hand or tremble, may choose unintended strategies.




$85 $95$85 500,500 1000,0$95 0,1000 800,800

Costco

Sam's

Comment 2: Sam’s and Costcoin the price competition gameboth gain by monopolizingor cartelizing the membership warehouse club industry and keeping prices high, but to do so requires playing a dominated strategy. The problem is that the group’s success in resolving their dilemma and fixing high prices harms the general public’s interest (as measured by total surplus). In the United States, the Sherman Antitrust Act prohibits such price or quantity fixing “in restraint of trade or commerce”. Violations can lead to jail terms for the firms’ executives, not just fines for the corporations.



$85 $95$85 500,500 1000,0$95 0,1000 800,800

Costco

Sam's

In the industry for large turbinesthat generate electricity,GE was the largest producerin the 1950s, with 60 percentof the market. Westinghouse has 30 percent, and Allied-Chambers had 10 percent. They kept those shares and obtained high prices though a cleaver coordination device.

Electric utilities invited bids for the turbines they intended to buy. If the bid was issued during days 1-17 of a lunar month, Westinghouse and Allied-Chambers had to put in very high bids that would be sure losers, and GE was the chosen winner. Similarly, Westinghouse was the chosen winner for days 18-25, and Allied-Chambers for days 26-28. Eventually the Department of Justice figured it out, and some executives went to jail.



$85 $95$85 500,500 1000,0$95 0,1000 800,800

Costco

Sam's

In the retail industry detectionof price cuts that violateprice-setting agreementsAnd the punishment of suchviolations can be simplified and retaliation made quick and automatic by low price guarantees.

At first sight, low price guarantees seem to guarantee low prices. But game-theoretic thinking shows that in reality they can have exactly the opposite effect.

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Question: Sam’s Club and Costco reconsider their price competition. As before, both sell emergency food supplies in a weather-proof bucket that provides 275 delicious easy-to-prepare meals, including potato soup and corn chowder. The unit cost to both retailers is $75. The retailers compete on price: the low-price retailer gets all the market and they split the market if they have equal prices. Suppose they simultaneously consider prices $75, $85, and $95, and suppose market demands at those prices are 120, 100, and 80.

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But now suppose each store offers the following low-price guarantee: “We guarantee lower prices than any other store, and we do everything in our power to ensure that you’re not paying too much for your purchase. That’s why we offer our Low Price Guarantee. If you find a lower advertised price, simply let us know and we’ll gladly meet that price!”

To decide the effect of that guarantee, redefine the normal form for the Price Competition Game modified by the Low Price Guarantee, and check for dominate strategies in that game, and compare with the original Price Competition Game.

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$75 $85 $95$75 0,0 0,0 0,0$85 0,0 500,500 1000,0$95 0,0 0,1000 800,800

Costco

Sam's

Answer: The Price Competition Game has a weakly dominate strategyfor each player:

Sam’s price = $85 gives at least as good payoffs for Sam’s compared with $75 or $95, no matter Costco’s price, and better payoffs if Costco picks $85.

Costco’s price = $85 gives at least as good payoffs for Costco compared with $75 or $95, no matter Sam’s price, and better payoffs if Sam’s picks $85.

Conclusion: Both choose $85.

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$75 $85 $95$75 0,0 0,0 0,0$85 0,0 500,500 500,500$95 0,0 500,500 800,800

Costco

Sam's


To define the normal form for the Modified Price Competition Game, at Sam’s price $95 and Costco price $85, Sam’s reduces price to $85 and splits the market demand of 100; hence, both make $(85-75)x50 = $500.

At Sam’s price $95 and Costco price $75, Sam’s reduces price to $75 and splits the market demand of 140; hence, both make $(75-75)x70 = $0.

At Sam’s price $85 and Costco price $75, Sam’s reduces price to $75 and splits the market demand of 140; hence, both make $(75-75)x70 = $0.

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$75 $85 $95$75 0,0 0,0 0,0$85 0,0 500,500 500,500$95 0,0 500,500 800,800

Costco

Sam's


The Price Competition Game modified by The Low Price Guarantee has a weakly dominate strategy for each player:

Sam’s price = $95 gives at least as good payoffs for Sam’s compared with $75 or $85, no matter Costco’s price, and better payoffs if Costco picks $95.

Costco’s price = $95 gives at least as good payoffs for Costco compared with $75 or $85, no matter Sam’s price, and better payoffs if Sam’s picks $95.

Conclusion: The “Low Price Guarantee” guarantees high prices.


Example 4: Dominated Strategies



Overview

Dominated Strategies for a player give worse payoffs compared with any other strategy, no matter what other players choose for their strategies. No rational player should choose a dominated strategy.



Comment: A dominated strategy for a player gives worse payoffs for that player compared with some other strategy, no matter what other players choose for their strategies. While dominate strategies are the recommended choice to play games, dominated strategies should never be chosen. Eliminating dominated strategies reduces the game, and the new game may have further dominated strategies, which can be eliminated, and so on.



Question: Woolworths and Coles control a large share of the Australian supermarket market. In January of next year, they are each planning to open a new store somewhere in Sydney, either The Central Business District (CBD) or King’s Cross. They face a tension between locating far apart, giving each some local market power, and locating near customers. That tension between monopoly power and competition results in the profit payoffs in the normal form below. Where should the stores locate?


CBD King'sCBD 3,4 1,3

King's 4,1 2,2

Coles

Wool

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CBD King'sCBD 3,4 1,3

King's 4,1 2,2

Coles

Wool


Answer: Coles has no dominate nor weakly dominate nor dominated nor weakly dominated strategies.But Woolworths has King’s Cross as dominate, and so CBD as dominated. After eliminating the latter, Coles now has King’s Cross as dominate.

Thus, the optimum for Woolworths is King’s Cross and the optimum for Coles is King’s Cross.

Comment: Those optima are for each individual player. If the players colluded, then CBD is better for both players. But that is hard to enforce since Woolworths would be playing a dominated strategy.



Example 5: Weakly Dominated Strategies



Overview

Weakly Dominated Strategies give at least as bad payoffs compared with any other strategy, no matter what other players choose, and worse payoffs for at least one choice of the other players.



Comment: A weakly dominated strategy for a player gives at least as bad payoffs for that player compared with some other strategy, no matter what other players choose for their strategies, and worse payoffs for at least one choice of strategies for the other players. Eliminating weakly-dominated strategies reduces the game, and the new game may have further weakly-dominated strategies, which can be eliminated, and so on.



Question: FedEx and UPS control a large share of the express small package delivery market. The unit cost to both firms is $75. The retailers compete on price: the low-price retailer gets all the market and they split the market if they have equal prices.



Suppose FedEx and UPS simultaneously consider prices $75, $76, $77, $78, $79, $80, $85, and $95, and suppose market demands at those prices are quantities 140, 136, 132, 128, 124, 120, 100, and 80. Fill in the empty cells of the profit table below, and find optimal prices.

11 22 75 76 77 78 79 80 85 9575 0,0 0,0 0,0 0,0 0,0 0,0 0,0 0,076 0,0 68,68 136,0 136,0 136,0 136,0 136,0 136,077 0,0 0,136 132,132 264,0 264,0 264,0 264,0 264,078 0,0 0,136 0,264 384,0 384,0 384,079 0,0 0,136 0,264 496,0 496,0 496,080 0,0 0,136 0,264 0,384 0,496 300,300 600,0 600,085 0,0 0,136 0,264 0,384 0,496 0,600 500,500 1000,095 0,0 0,136 0,264 0,384 0,496 0,600 0,1000 800,800

UPS

FedEx


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Step 1: For each firm, $75 is weakly dominated by any other strategy. Hence, eliminate $75 and reduce the game.

11 22 75 76 77 78 79 80 85 9575 0,0 0,0 0,0 0,0 0,0 0,0 0,0 0,076 0,0 68,68 136,0 136,0 136,0 136,0 136,0 136,077 0,0 0,136 132,132 264,0 264,0 264,0 264,0 264,078 0,0 0,136 0,264 192,192 384,0 384,0 384,0 384,079 0,0 0,136 0,264 0,384 248,248 496,0 496,0 496,080 0,0 0,136 0,264 0,384 0,496 300,300 600,0 600,085 0,0 0,136 0,264 0,384 0,496 0,600 500,500 1000,095 0,0 0,136 0,264 0,384 0,496 0,600 0,1000 800,800

UPS

FedEx

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Step 2: For each firm, $95 is weakly dominated by $85. Hence, eliminate $95 and reduce the game.

11 22 75 76 77 78 79 80 85 9575 0,0 0,0 0,0 0,0 0,0 0,0 0,0 0,076 0,0 68,68 136,0 136,0 136,0 136,0 136,0 136,077 0,0 0,136 132,132 264,0 264,0 264,0 264,0 264,078 0,0 0,136 0,264 192,192 384,0 384,0 384,0 384,079 0,0 0,136 0,264 0,384 248,248 496,0 496,0 496,080 0,0 0,136 0,264 0,384 0,496 300,300 600,0 600,085 0,0 0,136 0,264 0,384 0,496 0,600 500,500 1000,095 0,0 0,136 0,264 0,384 0,496 0,600 0,1000 800,800

UPS

FedEx

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Step 3: For each firm, $85 is weakly dominated by $80. Hence, eliminate $85 and further reduce the game.

11 22 75 76 77 78 79 80 85 9575 0,0 0,0 0,0 0,0 0,0 0,0 0,0 0,076 0,0 68,68 136,0 136,0 136,0 136,0 136,0 136,077 0,0 0,136 132,132 264,0 264,0 264,0 264,0 264,078 0,0 0,136 0,264 192,192 384,0 384,0 384,0 384,079 0,0 0,136 0,264 0,384 248,248 496,0 496,0 496,080 0,0 0,136 0,264 0,384 0,496 300,300 600,0 600,085 0,0 0,136 0,264 0,384 0,496 0,600 500,500 1000,095 0,0 0,136 0,264 0,384 0,496 0,600 0,1000 800,800

UPS

FedEx

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11 22 75 76 77 78 79 80 85 9575 0,0 0,0 0,0 0,0 0,0 0,0 0,0 0,076 0,0 68,68 136,0 136,0 136,0 136,0 136,0 136,077 0,0 0,136 132,132 264,0 264,0 264,0 264,0 264,078 0,0 0,136 0,264 192,192 384,0 384,0 384,0 384,079 0,0 0,136 0,264 0,384 248,248 496,0 496,0 496,080 0,0 0,136 0,264 0,384 0,496 300,300 600,0 600,085 0,0 0,136 0,264 0,384 0,496 0,600 500,500 1000,095 0,0 0,136 0,264 0,384 0,496 0,600 0,1000 800,800

UPS

FedEx

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11 22 75 76 77 78 79 80 85 9575 0,0 0,0 0,0 0,0 0,0 0,0 0,0 0,076 0,0 68,68 136,0 136,0 136,0 136,0 136,0 136,077 0,0 0,136 132,132 264,0 264,0 264,0 264,0 264,078 0,0 0,136 0,264 192,192 384,0 384,0 384,0 384,079 0,0 0,136 0,264 0,384 248,248 496,0 496,0 496,080 0,0 0,136 0,264 0,384 0,496 300,300 600,0 600,085 0,0 0,136 0,264 0,384 0,496 0,600 500,500 1000,095 0,0 0,136 0,264 0,384 0,496 0,600 0,1000 800,800

UPS

FedEx

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11 22 75 76 77 78 79 80 85 9575 0,0 0,0 0,0 0,0 0,0 0,0 0,0 0,076 0,0 68,68 136,0 136,0 136,0 136,0 136,0 136,077 0,0 0,136 132,132 264,0 264,0 264,0 264,0 264,078 0,0 0,136 0,264 192,192 384,0 384,0 384,0 384,079 0,0 0,136 0,264 0,384 248,248 496,0 496,0 496,080 0,0 0,136 0,264 0,384 0,496 300,300 600,0 600,085 0,0 0,136 0,264 0,384 0,496 0,600 500,500 1000,095 0,0 0,136 0,264 0,384 0,496 0,600 0,1000 800,800

UPS

FedEx

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Step 7: For each firm, $77 is weakly dominated by $76. Hence, eliminate $77 and solve the game with prices $76 for each firm.

11 22 75 76 77 78 79 80 85 9575 0,0 0,0 0,0 0,0 0,0 0,0 0,0 0,076 0,0 68,68 136,0 136,0 136,0 136,0 136,0 136,077 0,0 0,136 132,132 264,0 264,0 264,0 264,0 264,078 0,0 0,136 0,264 192,192 384,0 384,0 384,0 384,079 0,0 0,136 0,264 0,384 248,248 496,0 496,0 496,080 0,0 0,136 0,264 0,384 0,496 300,300 600,0 600,085 0,0 0,136 0,264 0,384 0,496 0,600 500,500 1000,095 0,0 0,136 0,264 0,384 0,496 0,600 0,1000 800,800

UPS

FedEx


Comment: The dominance solution where managers from each firm simultaneously charge price $76 barely above marginal cost is the result of each firm successively eliminating prices $95, $85, $80, $79, $78, and $77. It is like the affect of a sequential price war between two firms, but with the dominance solution, each firm simultaneously jumps right to the end and charges $76. An outside observer would thus never see the sequence of prices in the “price war” in the dominance solution because the “price war” and the sequence of eliminations of dominated strategies is in the minds of the managers.



Example 6: Rationalizable Strategies



Overview

Rationalizable Strategies are a best response to some belief about what other players choose, where the other players are, themselves, believed to choose rationalizable strategies.



Comment: Rationalizable strategy choices in a game can be justified purely on the basis of rationality and the common knowledge of rationality.


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11 22 C1 C2 C3 C4S1 1,7 2,5 0,0 1,1S2 1,2 0,3 1,2 2,1S3 2,0 3,5 2,7 0,1S4 0,0 0,-2 0,0 4,-1

Target

Wal-Mart


Question: Wal-Mart and Target control a large share of the SoCal discount department store market. In June of next year, they are each planning to open a new store somewhere in Los Angeles (Northridge, North Hollywood, Brentwood, or San Pedro). They face a tension between locating far apart, giving each some local market power, and locating near customers. That tension between monopoly power and competition results in the profit payoffs in the normal form below. Where should the stores locate?


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11 22 C1 C2 C3 C4S1 1,7 2,5 0,0 1,1S2 1,2 0,3 1,2 2,1S3 2,0 3,5 2,7 0,1S4 0,0 0,-2 0,0 4,-1

Target

Wal-Mart


Answer: There are no dominate strategies nor dominated strategies in the normal form. However, no matter what Target believes about Wal-Mart, Target would not choose location C4 as a best response. Likewise, no matter what Wal-Mart believes about Target, Wal-Mart would not choose locations S1 or S2 as a best response. For that reason, S1 and S2 and C4 are not rationalizable, and can thus be eliminated.



Answer: After those eliminations, S3 is now dominate for Wal-Mart, and C3 is Target’s best responseto S3.

Thus, the combination (S3,C3) is the dominance solution to the location game.

11 22 C1 C2 C3 C4S1 1,7 2,5 0,0 1,1S2 1,2 0,3 1,2 2,1S3 2,0 3,5 2,7 0,1S4 0,0 0,-2 0,0 4,-1

Target

Wal-Mart



Summary

Summary

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Summary


When your opponents’ strategies are chosen simultaneously with yours, choosing your optimal strategies and forming beliefs about your opponents’ strategies can be hard since your opponents are simultaneously forming beliefs about you.

There are 5 ways to choosing your optimal strategies and forming beliefs about your opponents’ strategies. And these can be used in any combination.

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Summary


Choose Dominate Strategies. Those are strategies for a player

that give better payoffs for that player compared with any other strategy, no matter what other players choose for their strategies.

Weakly Dominate Strategies. Those are strategies for a player that give at least as good payoffs for that player compared with any other strategy, no matter what other players choose for their strategies, and better payoffs for at least one choice of strategies for the other players.

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Summary


Eliminate Dominated Strategies. Those are strategies for a

player that give worse payoffs for that player compared with some other strategy, no matter what other players choose for their strategies.

Weakly Dominated Strategies. Those are strategies for a player that give at least as bad payoffs for that player compared with some other strategy, no matter what other players choose for their strategies, and worse payoffs for at least one choice of strategies for the other players.

Non-Rationalizable Strategies. Those are strategies for a player that are never a best response for that player no matter what that player believes the other players choose for their strategies.

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Summary


The dominance solution to a game is the unique result from a sequence of selecting Dominate Strategies or Weakly Dominate Strategies and of eliminating Dominated Strategies and Weakly Dominated Strategies and Non-Rationalizable Strategies.

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Review Questions


Review Questions You should try to answer some of the review questions

(see the online syllabus) before the next class. You will not turn in your answers, but students may

request to discuss their answers to begin the next class. Your upcoming Exam 2 and cumulative Final Exam will

contain some similar questions, so you should eventually consider every review question before taking your exams.

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End of Lesson B.4

BA 445 Managerial Economics


1 1 deep thought ba 445 lesson b.4 simultaneous price competition as the light changed from red to...

Documents

players beliefs

extent of players rationality

opponents strategies

optimal strategies

rationalizable strategies

weakly dominated strategies

dominated strategy

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