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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.)

BA 210 Lesson II.4 Simultaneous Price Competition

Lesson II.4 Simultaneous Price Competition

Example 1: Forming Beliefs

Example 2: A Normal Form

Example 3: Dominate Strategies

Example 4: Weakly Dominate Strategies

Example 5: Dominated Strategies

Example 6: Weakly Dominated Strategies

Example 7: Rationalizable Strategies

Summary

Review Questions

BA 210 Lesson II.4 Simultaneous Price Competition

Lesson 1 formulates and solves the following games:

Example 1: 2/3 of the Average Game. Has a simple solution.

Examples 2 and 3 and 4 and 6: Price Competition Game. Has a complex solution requiring a game table to predict current actions by the other players.

Example 5: Budget Balance Game.

Example 7: Location Game.

BA 210 Lesson II.4 Simultaneous Price Competition

Example 1: Forming Beliefs

BA 210 Lesson II.4 Simultaneous Price Competition

Comment: Sequential move games 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.

BA 210 Lesson II.4 Simultaneous Price Competition

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

Rules:

No talking or other communication between players.

Players secretly write a real number between 0 and 100.

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

What number should you guess?

Example 1: Forming Beliefs about Current Strategies

BA 210 Lesson II.4 Simultaneous Price Competition

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.

Example 1: Forming Beliefs about Current Strategies

BA 210 Lesson II.4 Simultaneous Price Competition

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.

Example 1: Forming Beliefs about Current Strategies

BA 210 Lesson II.4 Simultaneous Price Competition

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

Example 1: Forming Beliefs about Current Strategies

BA 210 Lesson II.4 Simultaneous Price Competition

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 1: Forming Beliefs about Current Strategies

BA 210 Lesson II.4 Simultaneous Price Competition

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.

BA 210 Lesson II.4 Simultaneous Price Competition

Question: Sam’s Club and Costco 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 consider prices $75, $85, and $95, and suppose market demands at those prices are 140, 100, and 80.

Define the normal form for this Price Competition Game.

BA 210 Lesson II.4 Simultaneous Price Competition

Answer: To begin, at Sam's Club price $95 and Costco price $85, Costco gets the entire market demand of 100. Hence, Sam's makes $0 and Costco makes $(85-75)x100 = $1,000.

BA 210 Lesson II.4 Simultaneous Price Competition

Example 3: Dominate Strategies

Example 3: Dominate Strategies

BA 210 Lesson II.4 Simultaneous Price Competition

Example 3: Dominate Strategies

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

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

BA 210 Lesson II.4 Simultaneous Price Competition

Example 3: Dominate Strategies

Question: Restrict Sam’s Club and Costco in the Price Competition Game to choose between prices $85 and $95, but 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 market if they have equal prices, and keep market demands at 100 and 80 for prices $85 and $95.

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

BA 210 Lesson II.4 Simultaneous Price Competition

Example 3: Dominate Strategies

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.

BA 210 Lesson II.4 Simultaneous Price Competition

Example 3: Dominate Strategies

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.

BA 210 Lesson II.4 Simultaneous Price Competition

Example 4: Weakly Dominate Strategies

Example 4: Weakly Dominate Strategies

BA 210 Lesson II.4 Simultaneous Price Competition

Example 4: Weakly Dominate Strategies

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.

BA 210 Lesson II.4 Simultaneous Price Competition

Example 4: Weakly Dominate Strategies

Comment 2: Sam’s and Costco

in the price competition game

both gain by monopolizing

or 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.

BA 210 Lesson II.4 Simultaneous Price Competition

Example 4: Weakly Dominate Strategies

In the industry for large turbines

that generate electricity,

GE was the largest producer

in the 1950s, with 60 percent

of 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.

BA 210 Lesson II.4 Simultaneous Price Competition

Example 4: Weakly Dominate Strategies

In the retail industry detection

of price cuts that violate

price-setting agreements

And the punishment of such

violations 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.

BA 210 Lesson II.4 Simultaneous Price Competition

Example 4: Weakly Dominate Strategies

Question: Sam’s Club and Costco consider modifying the price competition as described in Example 2 with 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, define the normal form for the Price Competition Game modified by the Low Price Guarantee, and check for dominate strategies in that game and in the original Price Competition Game.

BA 210 Lesson II.4 Simultaneous Price Competition

Example 4: Weakly Dominate Strategies

Answer: The Price Competition Game

has a weakly dominate strategy

for each player:

Sam’s price = $85gives 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.

BA 210 Lesson II.4 Simultaneous Price Competition

Example 4: Weakly Dominate Strategies

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.

BA 210 Lesson II.4 Simultaneous Price Competition

Example 4: Weakly Dominate Strategies

The Price Competition Game modified by

The Low Price Guarantee

has a weakly dominate strategy

for each player:

Sam’s price = $95gives 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.

BA 210 Lesson II.4 Simultaneous Price Competition

Example 5: Dominated Strategies

Example 5: Dominated Strategies

BA 210 Lesson II.4 Simultaneous Price Competition

Example 5: Dominated Strategies

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.

BA 210 Lesson II.4 Simultaneous Price Competition

Example 5: Dominated Strategies

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

BA 210 Lesson II.4 Simultaneous Price Competition

Example 5: Dominated Strategies

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

Find optimal strategies for Congress and the Fed.

BA 210 Lesson II.4 Simultaneous Price Competition

Example 5: Dominated Strategies

Answer: The Federal Reserve

has no dominate nor weakly

dominate nor dominated nor

weakly dominated strategies.

But Congress has Budget Deficit as dominate, and so Budget Balance as dominated. After eliminating the latter, the Federal Reserve now has High Rates as dominate.

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

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

BA 210 Lesson II.4 Simultaneous Price Competition

Example 6: Weakly Dominated Strategies

Example 6: Weakly Dominated Strategies

BA 210 Lesson II.4 Simultaneous Price Competition

Example 6: Weakly Dominated Strategies

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.

BA 210 Lesson II.4 Simultaneous Price Competition

Example 6: Weakly Dominated Strategies

Question: Modify the Price Competition Game between Sam’s Club and Costco by supposing they consider

prices $75, $79, $80, $85, and $95,

and suppose market demands at those prices are

quantities 140, 124, 120, 100, and 80. Keep unit cost = $75. Fill in the empty cells below, and find optimal prices.

BA 210 Lesson II.4 Simultaneous Price Competition

Example 6: Weakly Dominated Strategies

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

BA 210 Lesson II.4 Simultaneous Price Competition

Example 6: Weakly Dominated Strategies

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

BA 210 Lesson II.4 Simultaneous Price Competition

Example 6: Weakly Dominated Strategies

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

BA 210 Lesson II.4 Simultaneous Price Competition

Example 7: Rationalizable Strategies

Example 7: Rationalizable Strategies

BA 210 Lesson II.4 Simultaneous Price Competition

Example 7: Rationalizable Strategies

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

BA 210 Lesson II.4 Simultaneous Price Competition

Example 7: Rationalizable Strategies

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

BA 210 Lesson II.4 Simultaneous Price Competition

Example 7: Rationalizable Strategies

Answer: There are no dominate

strategies nor dominated

strategies in the normal form.

However, no matter what Costco

believes about Sam’s, Costco would not choose location C4 as a best response. Likewise, no matter what Sam’s believes about Costco, Sam’s 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.

BA 210 Lesson II.4 Simultaneous Price Competition

Example 7: Rationalizable Strategies

Answer: After those eliminations,

S3 is now dominate for Sam’s,

and C3 is Costco’s best response

to S3.

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

BA 210 Lesson II.4 Simultaneous Price Competition

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.

BA 210 Lesson II.4 Simultaneous Price Competition

- 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.

BA 210 Lesson II.4 Simultaneous Price Competition

- 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.

BA 210 Lesson II.4 Simultaneous Price Competition

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.

BA 210 Lesson II.4 Simultaneous Price Competition

- Review Questions
- You should try to answer some of the following questions 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.

BA 210 Lesson II.4 Simultaneous Price Competition

BA 210 Introduction to Microeconomics

End of Lesson II.4BA 210 Lesson II.4 Simultaneous Price Competition

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