mathematical analysis of hiroshima

8/10/2019 Mathematical Analysis of Hiroshima

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Fernando & Yvonn Quijano

Prepared by:

Game Theoryand Competitive

Strategy

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Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall Microeconomics Pindyck/Rubinfeld, 8e.


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2 of 42Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall Microeconomics Pindyck/Rubinfeld, 8e.

CHAPTER 13 OUTLINE

13.1 Gaming and Strategic Decisions

13.2 Dominant Strategies

13.3 The Nash Equilibrium Revisited

13.4 Repeated Games13.5 Sequential Games

13.6 Threats, Commitments, and Credibility

13.7 Entry Deterrence

13.8 Auctions


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GAMING AND STRATEGIC DECISIONS13.1

game Situation in which players

(participants) make strategic decisionsthat take into account each others actions

and responses.

payoff Value associated with a possible

outcome.

strategy Rule or plan of action for

playing a game.

optimal strategy Strategy that

maximizes a players expected payoff.

If I believe that my competitors are rational and act to maximize their

own payoffs, how should I take their behavior into account when making

my decisions?


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cooperative game Game in which

participants can negotiate bindingcontracts that allow them to plan jointstrategies.

noncooperative game Game in whichnegotiation and enforcement of bindingcontracts are not possible.

Noncooperative versus Cooperative Games

It is essential to understand your opponents point of view and to deduce

his or her likely responses to your actions.


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Noncooperative versus Cooperative Games

How to Buy a Dollar Bill

A dollar bill is auctioned, but in an unusual way. The highest bidderreceives the dollar in return for the amount bid.

However, the second-highest bidder must also hand over the amountthat he or she bidand get nothing in return.

If you were playing this game, how much would you bid for the dollar

bill?


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DOMINANT STRATEGIES13.2

dominant strategy Strategy that is

optimal no matter what an opponent does.

Suppose FirmsA and Bsell competing products and are decidingwhether to undertake advertising campaigns. Each firm will beaffected by its competitors decision.


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DOMINANT STRATEGIES13.2

equilibrium in dominant strategies

Outcome of a game in which each firm isdoing the best it can regardless of what itscompetitors are doing.

Unfortunately, not every game has a dominant strategy for each player. Tosee this, lets change our advertising example slightly.


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THE NASH EQUILIBRIUM REVISITED13.3

Dominant Strategies: Im doing the best I can no matter what you do.

Youre doing the best you can no matter what I do.

Nash Equilibrium: Im doing the best I can given what you are doing.

Youre doing the best you can given what I am doing.

The Product Choice Problem

Two breakfast cereal companies face a market in which two newvariations of cereal can be successfully introduced.


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The Beach Location Game

You (Y) and a competitor (C) plan to sell soft drinks on a beach.

If sunbathers are spread evenly across the beach and will walk to the closest vendor, the two of youwill locate next to each other at the center of the beach. This is the only Nash equilibrium.

If your competitor located at pointA, you would want to move until you were just to the left, where youcould capture three-fourths of all sales.

But your competitor would then want to move back to the center, and you would do the same.

Beach Location Game

Figure 13.1


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Maximin Strategies

The concept of a Nash equilibrium relies heavily on individualrationality. Each players choice of strategy depends not only on its

own rationality, but also on the rationality of its opponent. This can bea limitation.

cooperative game Game in which

participants can negotiate bindingcontracts that allow them to plan jointstrategies.


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Maximin Strategies

If Firm 1 is unsure about what Firm 2 will do but can assignprobabilities to each feasible action for Firm 2, it could insteaduse a strategy that maximizes its expected payoff.

Maximizing the Expected Payoff

The Prisoners Dilemma

What is the Nash equilibrium for the prisoners dilemma?


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Mixed Strategies

In this game, each player chooses heads or tails and the twoplayers reveal their coins at the same time. If the coins match

PlayerAwins and receives a dollar from Player B. If the coins donot match, Player Bwins and receives a dollar from PlayerA.

pure strategy Strategy in which a player makes a specificchoice or takes a specific action.

Matching Pennies

mixed strategy Strategy in which a player makes a random choice among

two or more possible actions, based on a set of chosen probabilities.


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Mixed Strategies

Jim and Joan would like to spend Saturday night together but havedifferent tastes in entertainment. Jim would like to go to the opera,but Joan prefers mud wrestling.

The Battle of the Sexes


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REPEATED GAMES13.4

How does repetition change the likely outcome of the game?

repeated game Game in which

actions are taken and payoffsreceived over and over again.


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REPEATED GAMES13.4

Suppose the game is infinitely repeated. In other words, mycompetitor and I repeatedly set prices month after month, forever.

With infinite repetition of the game, the expected gains fromcooperation will outweigh those from undercutting.

tit-for-tat strategy Repeated-game

strategy in which a player responds inkind to an opponents previous play,

cooperating with cooperativeopponents and retaliating against

uncooperative ones.

Tit-for-Tat Strategy

Infinitely Repeated Game


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REPEATED GAMES13.4

Finite Number of Repetitions

Now suppose the game is repeated a finitenumber of timessay, Nmonths.

Because Firm 1 is playing tit-for-tat, I (Firm 2) cannot undercutthat is,until the last month. I shouldundercut the last month because then I can

make a large profit that month, and afterward the game is over, so Firm 1cannot retaliate. Therefore, I will charge a high price until the last month,and then I will charge a low price.

However, since I (Firm 1) have also figured this out, I also plan to chargea low price in the last month. Firm 2 figures that it should undercut andcharge a low price in the next-to-last month.

And because the same reasoning applies to each preceding month, thegame unravels: The only rational outcome is for both of us to charge alow price every month.


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REPEATED GAMES13.4

Tit-for-Tat in Practice

Since most of us do not expect to live forever, the unraveling argumentwould seem to make the tit-for-tat strategy of little value, leaving us stuckin the prisoners dilemma. In practice, however, tit-for-tat can sometimeswork and cooperation can prevail.

There are two primary reasons.Most managers dont know how long they will be competing with

their rivals, and this also serves to make cooperative behavior agood strategy.

My competitor might have some doubt about the extent of my

rationality.

In a repeated game, the prisoners dilemma can have a cooperative

outcome.


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REPEATED GAMES13.4

Almost all the water meters sold in the United Stateshave been produced by four American companies.Rockwell International has had about a 35-percentshare of the market, and the other three firms havetogether had about a 50- to 55-percent share.

Most buyers of water meters are municipal water utilities, who install the metersin order to measure water consumption and bill consumers accordingly.

Utilities are concerned mainly that the meters be accurate and reliable. Price isnot a primary issue, and demand is very inelastic.

Because any new entrant will find it difficult to lure customers from existingfirms, this creates a barrier to entry. Substantial economies of scale create asecond barrier to entry.

The firms thus face a prisoners dilemma. Can cooperation prevail?

It can and hasprevailed. There is rarely an attempt to undercut price, and each

firm appears satisfied with its share of the market.


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REPEATED GAMES13.4

In March 1983, American Airlines proposed that all airlinesadopt a uniform fare schedule based on mileage. The rateper mile would depend on the length of the trip, with thelowest rate of 15 cents per mile for trips over 2500 milesand the highest rate, 53 cents per mile, for trips under 250

miles.Why did American propose this plan, and what made it so attractive to the otherairlines?

The aim was to reduce price competition and achieve a collusive pricingarrangement. Fixing prices illegal. Instead, the companies would implicitly fix

prices by agreeing to use the same fare-setting formula.The plan failed, a victim of the prisoners dilemma.

Pan Am, which was dissatisfied with its small share of the U.S. market, dropped itsfares. American, United, and TWA, afraid of losing their own shares of the market,quickly dropped their fares to match Pan Am. The price-cutting continued, and

fortunately for consumers, the plan was soon dead.


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SEQUENTIAL GAMES13.5

As a simple example, lets return to the product choice

problem. This time, lets change the payoff matrix slightly.

sequential game Game in which

players move in turn, responding toeach others actions and reactions.


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extensive form of a gameRepresentation of possible moves ina game in the form of a decision tree.

The Extensive Form of a Game

Product Choice Game in Extensive Form

Figure 13.2


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The Advantage of Moving First


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THREATS, COMMITMENTS, AND CREDIBILITY13.6

Suppose Firm 1 produces personal computers that canbe used both as word processors and to do other tasks.Firm 2 produces only dedicated word processors.

Empty Threats


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Race Car Motors, Inc., produces cars, and Far Out Engines, Ltd.,produces specialty car engines.

Far Out Engines sells most of its engines to Race Car Motors, and afew to a limited outside market.

Nonetheless, it depends heavily on Race Car Motors and makes itsproduction decisions in response to Race Cars production plans.

Commitment and Credibility


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Suppose Far Out threatensto produce big engines no matter what RaceCar does. If Race Car believed Far Outs threat, it would produce big

cars: Otherwise, it would have trouble finding engines for its small cars.

Far Out can make its threat credible by visibly and irreversibly reducingsome of its own payoffs in the matrix, thereby constraining its own

choices.Far Out must reduce its profits from small engines. It might do this byshutting down or destroying some of its small engine production capacity.



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Developing the right kind of reputationcan also give one a strategicadvantage.

Suppose that the managers of Far Out Engines develop a reputationfor being irrationalperhaps downright crazy.

They threaten to produce big engines no matter what Race CarMotors does.

Now the threat might be credible without any further action; after all,you cant be sure that an irrational manager will always make a profit-maximizing decision.

In gaming situations, the party that is known (or thought) to be a littlecrazy can have a significant advantage.


The Role of Reputation


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Our discussion of commitment and credibility also applies tobargaining problems. The outcome of a bargaining situation can dependon the ability of either side to take an action that alters its relativebargaining position.

Consider two firms that are each planning to introduce one of two

products which are complementary goods.

Bargaining Strategy


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Suppose that Firms 1 and 2 are also bargaining over a second issuewhether to join a research consortium that a third firm is trying to form

Bargaining Strategy


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How did Wal-Mart Stores succeed whereothers failed? The key was Wal-Marts expansionstrategy.

The conventional wisdom held that a discount store could succeed only in a city

with a population of 100,000 or more. Sam Walton disagreed and decided toopen his stores in small Southwestern towns.

The stores succeeded because Wal-Mart had created local monopolies.Discount stores that had opened in larger cities were competing with otherdiscount stores. Other discount chains realized that Wal-Mart had a profitablestrategy, so the issue became who would get to each town first. Wal-Mart now

found itself in apreemption game.


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ENTRY DETERRENCE13.7

To deter entry, the incumbent firm must convince any potentialcompetitor that entry will be unprofitable.

Empty Threats

But what if you can make an irrevocable commitment that willalter your incentives once entry occursa commitment that willgive you little choice but to charge a low price if entry occurs?


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Strategic Trade Policy and International Competition

The development and production of a new line of aircraft aresubject to substantial economies of scale; it would not pay todevelop a new aircraft unless a firm expected to sell many of them.

Suppose it is only economical for one firm to produce the new

aircraft.

The Commercial Aircraft Market


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Strategic Trade Policy and International Competition

The Commercial Aircraft MarketEuropean governments, of course, would prefer that Airbus produce thenew aircraft. Can they change the outcome of this game?

Suppose they commit to subsidizing Airbus and make this commitmentbefore Boeing has committed itself to produce. If the Europeangovernments commit to a subsidy of 20 to Airbus if it produces theplane regardless of what Boeing does, the payoff matrix would change.


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In the early 1970s, DuPont and National Lead each accounted for about a thirdof U.S. titanium dioxide sales; another seven firms produced the remainder.DuPont was considering whether to expand capacity. The industry waschanging, and those changes might enable DuPont to capture more of themarket and dominate the industry

Three factors had to be considered:

Future demand was expected to grow substantially.

New environmental regulations would be imposed.

The prices of raw materials used to make titanium dioxide were rising.

The new regulations and the higher input prices would have a major effect onproduction cost and give DuPont a cost advantage, both because its productiontechnology was less sensitive to the change in input prices and because itsplants were in areas that made disposal of corrosive wastes much less difficultthan for other producers.


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DuPont anticipated that other producers would have to shut down part of theircapacity.

Competitors would in effect have to reenter the market by building new

plants. Could DuPont deter them from taking this step?

DuPont considered the strategy to invest nearly $400 million in increased

production capacity to try to capture 64 percent of the market by 1985.

The idea was to deter competitors from investing. Scale economies andmovement down the learning curve would give DuPont a cost advantage.

By 1975, things began to go awry.

Because demand grew by much less than expected, there was excesscapacity industrywide.

Because the environmental regulations were only weakly enforced,competitors did not have to shut down capacity as expected.

DuPonts strategy led to antitrust action by the Federal Trade Commission in

1978.


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The disposable diaper industry in the United States hasbeen dominated by two firms: Procter & Gamble, with anapproximately 50-percent market share, and Kimberly-Clark, with another 3040 percent.

How do these firms compete? And why havent other firms been able to enter

and take a significant share of this $5-billion-per-year market?

The competition occurs mostly in the form of cost-reducing innovation. As aresult, both firms are forced to spend heavily on research and development in arace to reduce cost.


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AUCTIONS13.8 auction market Market in which

products are bought and sold

through formal bidding processes.Auction Formats

English (ororal) auction Auction in which a

seller actively solicits progressively higher bidsfrom a group of potential buyers.

Dutch auction Auction in which a seller beginsby offering an item at a relatively high price, thenreduces it by fixed amounts until the item is sold.

sealed-bid auction Auction in which all bids are

made simultaneously in sealed envelopes, thewinning bidder being the individual who hassubmitted the highest bid.

first-price auction Auction in which the sales

price is equal to the highest bid.

second-price auction Auction in which the sales

price is equal to the second-highest bid.


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AUCTIONS13.8

Valuation and Information

private-value auction Auction in which each bidder

knows his or her individual valuation of the object upfor bid, with valuations differing from bidder to bidder.

common-value auction Auction in which the item

has the same value to all bidders, but bidders do notknow that value precisely and their estimates of it vary.

Private-Value Auctions

Whatever the auction format, each bidder must choose his or her biddingstrategy.

For an open English auction, this strategy is a choice of a price at

which to stop bidding.

For a Dutch auction, the strategy is the price at which the individualexpects to make his or her only bid.

For a sealed-bid auction, the strategy is the choice of bid to place ina sealed envelope.


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AUCTIONS13.8

Common-Value Auctions

Suppose that you and four other people participate in an oral auctionto purchase a large jar of pennies, which will go to the winning bidderat a price equal to the highest bid.

Once you have estimated the number of pennies in the jar, what isyour optimal bidding strategy?

The Winners Curse

winners curse Situation in which the winner

of a common-value auction is worse off as aconsequence of overestimating the value of the

item and thereby overbidding.


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AUCTIONS13.8

Maximizing Auction Revenue

Here are some useful tips for choosing the best auction format.

1. In a private-value auction, you should encourage as many bidders aspossible.

2. In a common-value auction, you should (a) use an open rather than asealed-bid auction because, as a general rule, an English (open) common-

value auction will generate greater expected revenue than a sealed-bidauction; and (b) reveal information about the true value of the object beingauctioned.

3. In a private-value auction, set a minimum bid equal to or even somewhathigher than the value to you of keeping the good for future sale.

Bidding and Collusion

Buyers can increase their bargaining power by reducing the number of biddersor the frequency of bidding. In some cases this can be accomplished legallythrough the formation of buying groups, but it may also be accomplishedillegally through collusive agreements that violate the antitrust laws.


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After Sothebys and Christies auction houses were found guilty in2001 of fixing commission prices, a federal class-action lawsuit followed.

When federal courts manage class-action suits, they are responsible forawarding attorneys fees. In this case, the judge decided to hold an auction to

select the law firm that would represent the plaintiff class.

The judge entertained secret sealed bids from 20 law firms. Each firm was told tooffer a fee arrangement consisting of a base and a percentage. A settlement ortrial award at or below the base would be given entirely to the plaintiffs, with thelaw firm receiving nothing. If the settlement or award was higher than the base,the law firm would receive the stated percentage of the amount over the base.

The winning bidder was the law firm of Boies, Schiller, & Flexner, which bid abase of $405 million and a percentage of 25 percent. Some suggested that thefirm might not work hard in the plaintiffs interest because the minimum might be

unachievable.

Boies settled with defendants for $512 million, earning the attorneys a $26.75million fee and generating just over $475 million for the class members.

AUCTIONS13.8


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How has eBay come to dominate the Internet auctionmarket? Why havent other Internet auction sites (such as

Yahoo and Amazon) succeeded in taking market sharefrom eBay?

The answer is that Internet auctions are subject to very

strong network externalities.

If you wanted to auction off some rare coins or Pokmon cards, which auctionsite would you choose?

The one that had the largest number of potential bidders.

Likewise, if you wanted to bid for rare coins or Pokmon cards, you wouldchoose the auction site with the largest number of sellers.

Because eBay was the first major Internet auction site, it began with a largemarket share, and its share grew thanks to the network externality.

AUCTIONS13.8

mathematical analysis of hiroshima

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