1 1 Deep Thought BA 210 Lesson II.4 Simultaneous Price Competition As the light changed from red to...

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Deep ThoughtDeep Thought

BA 210 Lesson II.4 Simultaneous Price CompetitionBA 210 Lesson II.4 Simultaneous Price Competition

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

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

2 2BA 210 Lesson II.4 Simultaneous Price CompetitionBA 210 Lesson II.4 Simultaneous Price Competition

OverviewOverview

OverviewOverview

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Lesson OverviewLesson Overview

BA 210 Lesson II.4 Simultaneous Price Competition

Lesson II.4 Simultaneous Price CompetitionLesson II.4 Simultaneous Price CompetitionExample 1: Forming BeliefsExample 1: Forming BeliefsExample 2: A Normal FormExample 2: A Normal FormExample 3: Dominate StrategiesExample 3: Dominate StrategiesExample 4: Weakly Dominate StrategiesExample 4: Weakly Dominate StrategiesExample 5: Dominated StrategiesExample 5: Dominated StrategiesExample 6: Weakly Dominated StrategiesExample 6: Weakly Dominated StrategiesExample 7: Rationalizable StrategiesExample 7: Rationalizable StrategiesSummarySummaryReview QuestionsReview Questions

4 4BA 210 Lesson II.4 Simultaneous Price Competition

Lesson 1 formulates and solves Lesson 1 formulates and solves the following games:the following games:Example 1: Example 1: 2/3 of the Average Game.2/3 of the Average Game. Has a simple solution. Has a simple solution.Examples 2 and 3 and 4 and 6: Examples 2 and 3 and 4 and 6: Price Competition Game.Price Competition Game. Has a Has a

complex solution requiring a game table to predict current complex solution requiring a game table to predict current actions by the other players.actions by the other players.

Example 5: Example 5: Budget Balance Game.Budget Balance Game. Example 7: Example 7: Location Game.Location Game.

Lesson OverviewLesson Overview


Example 1: Forming BeliefsExample 1: Forming Beliefs




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

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

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

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

Rules:Rules:1.1.No talking or other communication between players.No talking or other communication between players.2.2.Players secretly write a real number between 0 and 100.Players secretly write a real number between 0 and 100.3.3.The winner is the one closest to 2/3 of the average.The winner is the one closest to 2/3 of the average.

What number should you guess? What number should you guess?


Example 1: Forming Beliefs about Current StrategiesExample 1: Forming Beliefs about Current Strategies

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

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



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

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

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



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



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

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

BecauseBecause you assume other players are rational, you assume other players are rational, eliminate any eliminate any guess higher than guess higher than 44.45. 44.45.

Because you assume other players assume other players are Because you assume other players assume other players are rational, rational, eliminate any guess higher than eliminate any guess higher than 29.64. 29.64.

Without further assumptions, all guesses Without further assumptions, all guesses from 0 to from 0 to 29.64 are 29.64 are viable.viable.




Example 2: A Normal FormExample 2: A Normal Form



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



Question: Sam’s Club and Costco Question: Sam’s Club and Costco both sell emergency food both sell emergency food supplies in a weather-proof bucket that provides 275 delicious supplies in a weather-proof bucket that provides 275 delicious easy-to-prepare meals, including potato soup and corn chowder. easy-to-prepare meals, including potato soup and corn chowder. The unit cost to both retailers is $75. The retailers compete on 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 price: the low-price retailer gets all the market and they split the market if they have equal prices. Suppose they consider prices market if they have equal prices. Suppose they consider prices $75, $85, and $95, and suppose market demands at those prices $75, $85, and $95, and suppose market demands at those prices are 140, 100, and 80.are 140, 100, and 80.

Define the normal form for this Define the normal form for this Price Competition GamePrice Competition Game..



Answer: Answer: To begin, at Sam's Club price $95 and Costco price $85, To begin, at Sam's Club price $95 and Costco price $85, Costco gets the entire market demand of 100. Hence, Sam's Costco gets the entire market demand of 100. Hence, Sam's makes $0 and Costco makes $(85-75)makes $0 and Costco makes $(85-75)xx100 = $1,000.100 = $1,000.


$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


Example 3: Dominate StrategiesExample 3: Dominate Strategies



Comment: The simplest Comment: The simplest simultaneous move games to solve do simultaneous move games to solve do not require you to predict what your opponent will do now since not require you to predict what your opponent will do now since your best response is the same your best response is the same no matter what you believe other no matter what you believe other players choose for their strategies. players choose for their strategies.

A dominate strategy A dominate strategy for a player gives better payoffs for that for a player gives better payoffs for that player compared with any other strategy, no matter what other player compared with any other strategy, no matter what other players choose for their strategies. Any rational player should players choose for their strategies. Any rational player should choose a dominate strategy.choose a dominate strategy.



Question: Restrict Sam’s Club and Costco Question: Restrict Sam’s Club and Costco in the in the Price Price Competition Game to choose between prices Competition Game to choose between prices $85 and $95, but $85 and $95, but keep the unit cost to both retailers at $75, keep the assumption keep the unit cost to both retailers at $75, keep the assumption that the low-price retailer gets all the market and they split the that the low-price retailer gets all the market and they split the market if they have equal prices, and keep market demands at market if they have equal prices, and keep market demands at 100 and 80 for prices $85 and $95.100 and 80 for prices $85 and $95.

Define the normal form for this reduced Define the normal form for this reduced Price Competition Price Competition GameGame, and find optimal strategies., and find optimal strategies.



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

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

Costco

Sam's



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

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

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


Example 4: Weakly Dominate Example 4: Weakly Dominate StrategiesStrategies

Example 4: Weakly Dominate StrategiesExample 4: Weakly Dominate Strategies


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




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

Costco

Sam's

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



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

Costco

Sam's

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

Electric utilities invited bids for the turbines they intended to Electric utilities invited bids for the turbines they intended to buy. If the bid was issued during days 1-17 of a lunar month, 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 Westinghouse and Allied-Chambers had to put in very high bids that would be sure losers, and GE was the chosen winner. that would be sure losers, and GE was the chosen winner. Similarly, Westinghouse was the chosen winner for days 18-25, Similarly, Westinghouse was the chosen winner for days 18-25, and Allied-Chambers for days 26-28. Eventually the Department and Allied-Chambers for days 26-28. Eventually the Department of Justice figured it out, and some executives went to jail.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 In the retail industry detectiondetectionof price cuts that violateof price cuts that violateprice-setting agreementsprice-setting agreementsAnd the punishment of suchAnd the punishment of suchviolations can be simplified and retaliation made quick and violations can be simplified and retaliation made quick and automatic by low price guarantees. automatic by low price guarantees.

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


Question: Sam’s Club and Costco Question: Sam’s Club and Costco consider modifying the price consider modifying the price competition as described in Example 2 with the following low-competition as described in Example 2 with the following low-price guarantee: “We guarantee lower prices than any other store, price guarantee: “We guarantee lower prices than any other store, and we do everything in our power to ensure that you’re not 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 paying too much for your purchase. That’s why we offer our Low Price Guarantee. If you find a lower advertised price, simply let Price Guarantee. If you find a lower advertised price, simply let us know and we’ll gladly meet that price!”us know and we’ll gladly meet that price!”

To decide the effect of that To decide the effect of that guarantee, guarantee, define define ththe normal form for e normal form for the the Price Competition Game modified by the Price Competition Game modified by the Low Price Low Price GuaranteeGuarantee, and , and check for dominate strategies in that game and in check for dominate strategies in that game and in the original the original Price Competition Game.Price Competition Game.



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

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

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

Conclusion: Both choose $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 thTo define the normal form e normal form for the for the Modified Price Competition Modified Price Competition GameGame, at Sam’s price $95 and , at Sam’s price $95 and Costco price $85, Sam’s reduces Costco price $85, Sam’s reduces price to $85 and splits the market demand of 100; hence, both price to $85 and splits the market demand of 100; hence, both make $(85make $(8575)75)xx50 = $500.50 = $500.

At At Sam’s price $95 and Costco price $75, Sam’s reduces price to 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 and splits the market demand of 140; hence, both make $(75$(7575)75)xx70 = $0.70 = $0.

At At Sam’s price $85 and Costco price $75, Sam’s reduces price to 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 and splits the market demand of 140; hence, both make $(75$(7575)75)xx70 = $0.70 = $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 The Price Competition Game modified by Price Competition Game modified by The The Low Price Guarantee Low Price Guarantee has a weakly dominate strategyhas a weakly dominate strategyfor each player: for each player:

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

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

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



Example 5: Dominated StrategiesExample 5: Dominated Strategies



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



Question: Congress Question: Congress is under pressure to lower taxes and raise is under pressure to lower taxes and raise spending and, thereby, run a budget deficit. The Federal spending and, thereby, run a budget deficit. The Federal Reserve’s primary task is to prevent inflation, but it is also under Reserve’s primary task is to prevent inflation, but it is also under pressure to lower interest rates. The Fed prefers lower rates but pressure to lower interest rates. The Fed prefers lower rates but only if inflation is not a treat, such as when Congress balances its only if inflation is not a treat, such as when Congress balances its budget.budget.



Define the normal form Define the normal form for a simultaneous move game between for a simultaneous move game between Congress and the Fed. Congress likes best (payoff 4) a budget Congress and the Fed. Congress likes best (payoff 4) a budget deficit and low rates, next (payoff 3) budget balance and low deficit and low rates, next (payoff 3) budget balance and low rates, next (2) a budget deficit and high rates, and worst (1) rates, next (2) a budget deficit and high rates, and worst (1) budget balance and high rates. The Fed likes best (payoff 4) budget balance and high rates. The Fed likes best (payoff 4) budget balance and low rates, next (payoff 3) budget balance and budget balance and low rates, next (payoff 3) budget balance and high rates, next (2) a budget deficit and high rates, and worst (1) high rates, next (2) a budget deficit and high rates, and worst (1) a budget deficit and low rates.a budget deficit and low rates.

Find optimal strategies for Congress and the Fed. Find optimal strategies for Congress and the Fed.



Answer:Answer: The Federal Reserve The Federal Reservehas no dominate nor weaklyhas no dominate nor weaklydominate nor dominated nor dominate nor dominated nor weakly dominated strategies.weakly dominated strategies.But Congress has Budget Deficit as dominate, and so Budget But Congress has Budget Deficit as dominate, and so Budget Balance as dominated. After eliminating the latter, the Federal Balance as dominated. After eliminating the latter, the Federal Reserve now has High Rates as dominate.Reserve now has High Rates as dominate.

Thus, the optimum for Congress is Budget Deficit and the Thus, the optimum for Congress is Budget Deficit and the optimum for Federal Reserve is High Rates. optimum for Federal Reserve is High Rates.

Comment: Comment: Those optima are for each individual player. If the Those optima are for each individual player. If the players colluded, then Budget Balance and Low Interest Rates players colluded, then Budget Balance and Low Interest Rates are better for both players. But that is hard to enforce since are better for both players. But that is hard to enforce since Congress would be playing a dominated strategy.Congress would be playing a dominated strategy.


Low RatesHigh RatesB. Balance 3,4 1,3B. Deficit 4,1 2,2

Federal Reserve

Congress


Example 6: Weakly Dominated Example 6: Weakly Dominated StrategiesStrategies

Example 6: Weakly Dominated StrategiesExample 6: Weakly Dominated Strategies


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



Question: Modify Question: Modify the the Price Competition Game between Price Competition Game between Sam’s Sam’s Club and Costco by supposing they consider Club and Costco by supposing they consider prices $75, $79, $80, $85, and $95, prices $75, $79, $80, $85, and $95, and suppose market demands at those prices areand suppose market demands at those prices arequantities 140, 124, 120, 100, and 80. Keep unit cost = $75. Fill quantities 140, 124, 120, 100, and 80. Keep unit cost = $75. Fill in the empty cells below, and find optimal prices.in the empty cells below, and find optimal prices.


11 22 75 79 80 85 9575 0,0 0,0 0,0 0,0 0,079 0,0 ?,? 496,0 496,0 496,080 0,0 ?,? 300,300 600,0 600,085 0,0 0,496 0,600 500,500 1000,095 0,0 0,496 0,600 0,1000 800,800

Costco

Sam's


Step 1: Step 1: For each firm, $75 is weakly dominated by any other For each firm, $75 is weakly dominated by any other strategy, and $95 is weakly dominated by $85. Hence, estrategy, and $95 is weakly dominated by $85. Hence, eliminate liminate $75$75 and and $95 a$95 and reduce the game.nd reduce the game.


11 22 75 79 80 85 9575 0,0 0,0 0,0 0,0 0,079 0,0 248,248 496,0 496,0 496,080 0,0 0,496 300,300 600,0 600,085 0,0 0,496 0,600 500,500 1000,095 0,0 0,496 0,600 0,1000 800,800

Costco

Sam's


Step 2: Step 2: For each firm, $85 is now weakly dominated by $80. For each firm, $85 is now weakly dominated by $80. Hence, eHence, eliminate liminate $85 a$85 and further reduce the game.nd further reduce the game.


11 22 75 79 80 85 9575 0,0 0,0 0,0 0,0 0,079 0,0 248,248 496,0 496,0 496,080 0,0 0,496 300,300 600,0 600,085 0,0 0,496 0,600 500,500 1000,095 0,0 0,496 0,600 0,1000 800,800

Costco

Sam's


Step 3: Step 3: For each firm, $80 is now weakly dominated by $79. For each firm, $80 is now weakly dominated by $79. Hence, eHence, eliminate liminate $80 a$80 and further reduce the game to its single nd further reduce the game to its single solution of prices $79 for each firm.solution of prices $79 for each firm.


11 22 75 79 80 85 9575 0,0 0,0 0,0 0,0 0,079 0,0 248,248 496,0 496,0 496,080 0,0 0,496 300,300 600,0 600,085 0,0 0,496 0,600 500,500 1000,095 0,0 0,496 0,600 0,1000 800,800

Costco

Sam's


Example 7: Rationalizable StrategiesExample 7: Rationalizable Strategies



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

Costco

Sam's


Question: Sam’s Club and Costco Question: Sam’s Club and Costco are each planning to open a are each planning to open a new store somewhere in Los Angeles (Northridge, North new store somewhere in Los Angeles (Northridge, North Hollywood, Brentwood, or San Pedro) in January of next year. Hollywood, Brentwood, or San Pedro) in January of next year. They face a tension between locating far apart, giving each some They face a tension between locating far apart, giving each some local market power, and locating where more customers live. local market power, and locating where more customers live. That tension between monopoly power and competition results in That tension between monopoly power and competition results in the profit payoffs in the normal form below. Where should the the profit payoffs in the normal form below. Where should the stores locate? 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

Costco

Sam's


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



Answer: After those eliminations,Answer: After those eliminations, S3 is now dominate for Sam’s, S3 is now dominate for Sam’s, and C3 is Costco’s best responseand C3 is Costco’s best responseto S3.to S3.

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

Costco

Sam's



SummarySummary

SummarySummary

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SummarySummary


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

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

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SummarySummary


ChooseChoose Dominate Strategies. Dominate Strategies. Those are strategies Those are strategies for a player that give for a player that give

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

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

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SummarySummary


EliminateEliminate Dominated Strategies. Dominated Strategies. Those are strategies Those are strategies for a player that for a player that

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

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

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

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SummarySummary


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

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Review QuestionsReview Questions


Review QuestionsReview Questions You should try to answer some of the following questions You should try to answer some of the following questions before the next class. before the next class. You will not turn in your answers, but students may request You will not turn in your answers, but students may request to discuss their answers to begin the next class. to discuss their answers to begin the next class. Your upcoming Exam 2 and cumulative Final Exam will Your upcoming Exam 2 and cumulative Final Exam will contain some similar questions, so you should eventually contain some similar questions, so you should eventually consider every review question before taking your exams.consider every review question before taking your exams.

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End of Lesson II.4End of Lesson II.4


BA 210 Introduction to BA 210 Introduction to MicroeconomicsMicroeconomics

1 1 Deep Thought BA 210 Lesson II.4 Simultaneous Price Competition As the light changed from red to...

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