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Chapter 10: Problem 11

Suppose that the speed limit is 70 miles per hour on the freeway and that \(n\) drivers simultaneously and independently choose speeds from 70 to 100 . Everyone prefers to go as fast as possible, other things equal, but the police ticket any driver whose speed is strictly faster than the speeds of a fraction \(x\) of the other drivers, where \(x\) is a parameter such that \(0 \leq x \leq 1\). More precisely, for a given driver, let \(m\) denote the number of drivers that choose strictly lower speeds; then, such a driver is ticketed if and only if \(m /(n-1)>x\). Note that by driving 70, a driver can be sure that he will not be ticketed. Suppose the cost of being ticketed outweighs any benefit of going faster. (a) Model this situation as a noncooperative game by describing the strategy space and payoff function of an individual player. (b) Identify the Nash equilibria as best you can. Are there any equilibria in which the drivers choose the same speed? Are there any equilibria in which the drivers choose different speeds? How does the set of Nash equilibria depend on \(x\) ? (c) What are the Nash equilibria under the assumption that the police do not ticket anyone? (d) What are the Nash equilibria under the assumption that the police ticket everyone who travels more than 70 ? (e) If the same drivers play this game repeatedly, observing the outcome after each play, and there is some noisiness in their choices of speed, how would you expect their speeds to change over time as they learn to predict each other's speeds when \(x\) is near 100 and when \(x\) is near 0 ? Explain your intuition.

Short Answer

Expert verified
Nash equilibria vary with ticket rules: all speeds equal or adaptively different depending on \(x\). Without tickets, all speed 100; with all-over-70 tickets, all speed 70.

Step by step solution

01

Identify the Strategy Space

Each driver can choose a speed between 70 and 100 mph. This range of speeds represents the strategy space for each player in the game.
02

Define the Payoff Function

The payoff to each driver depends on their chosen speed and whether they receive a ticket. Drivers prefer faster speeds but face a cost if ticketed. If a driver chooses a speed higher than the speeds of more than a fraction \(x\) of the other drivers, they get ticketed, which incurs a negative payoff that outweighs the benefit of speed.
03

Nash Equilibria Analysis

In a Nash equilibrium, no player can benefit from changing their speed while others keep theirs constant. If all drivers choose the same speed (game remains constant with choice), then that speed is a Nash equilibrium. Different speeds can form an equilibrium if no driver chooses a speed that would lead to getting ticketed, adhering to the \(m/(n-1)>x\) rule. The set of Nash equilibria may vary depending on \(x\) because high \(x\) values allow more drivers to go at higher speeds without being ticketed.
04

Analyze with No Tickets Issued

When tickets are not issued to any driver, the game is trivial since the primary constraint is removed. Every driver will choose 100 mph, as it is the preferable speed, creating a single Nash equilibrium where all players go full speed.
05

Analyze When All Speeds Over 70 Are Ticketed

Given that any choice above 70 results in a ticket, every driver will choose a speed of 70 mph to avoid a negative payoff. Thus, the Nash equilibrium is for all drivers to choose 70 mph.
06

Expectation with Repeated Games and Learning

When drivers play the game repeatedly, they learn from past outcomes and may adjust their expectations about others’ speeds. With \(x\) near 100, drivers will eventually converge towards 100 mph as risk of ticketing reduces. With \(x\) near 0, drivers will learn to converge towards lower speeds, possibly the limit, to avoid tickets, since going faster increases their chances of being among the 'fast ticketed' individuals.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Noncooperative Game
In game theory, a noncooperative game is a framework where players make decisions independently without forming alliances or agreements. The essence of noncooperative games lies in the concept that every player acts to maximize their own utility. There's no collaboration, and each player is in it for themselves.
  • Each driver's decision about which speed to choose on the freeway can be considered a move in this noncooperative game.
  • The drivers cannot communicate or form alliances to decide on speeds together; they operate independently based on personal preferences and strategies.
  • The lack of a cooperative element leads each driver to consider only their own benefit and potential costs, such as receiving a ticket.
Understanding noncooperative games is crucial as they often model real-world competitive scenarios where no external enforcement of cooperative behavior exists.
Nash Equilibrium
Nash equilibrium is a key concept in game theory, referring to a situation where no player can benefit by changing their strategy while the other players keep theirs constant.
  • In the scenario with freeway drivers, a Nash equilibrium occurs when each driver has selected a speed such that no single driver gains by changing their speed alone.
  • If all drivers go to the same speed and no one gets ticketed, this forms a Nash equilibrium. Changing speed for one driver would result in a ticket or higher costs.
  • The Nash equilibrium can also consist of different speeds if no driver risks getting ticketed by the police, adhering to the condition \( m/(n-1) > x \).
The variability of Nash equilibria in this game highly depends on values of \( x \). A higher \( x \) allows more drivers to increase speed without being ticketed, leading to multiple possible equilibria.
Strategy Space
The strategy space in a game is defined by the set of all possible actions that players can choose from. For the speeding drivers,
  • The strategy space comprises choosing any speed between 70 and 100 mph.
  • This wide range reflects different strategies players might adopt given the rules and their inclination to avoid tickets.
  • The choice of 70 mph is included, being the lowest speed that guarantees no ticket under any circumstances.
Understanding the strategy space helps define how players navigate choices within given constraints, determining their approach towards an optimal solution or Nash equilibrium.
Payoff Function
The payoff function in this context refers to how each driver's choice translates into rewards or penalties, based on speeding versus getting ticketed.
  • Each driver wants to drive as fast as possible but faces a significant cost if ticketed.
  • The payoff function effectively balances the desire for speed against the risk and consequences of law enforcement.
  • Therefore, drivers' decisions are influenced not just by the speed but also by the comparative choices of other drivers, making the game complex and strategic.
Clearly defining the payoff function helps players assess potential risks and make better-informed choices, crucial in noncooperative settings.
Repeated Games
Repeated games occur when the same game is played by the same participants multiple times. Participants can learn from past outcomes and modify their strategies accordingly.
  • As drivers repeatedly play this game, they gain insights into other drivers' behaviors and speed tendencies.
  • The outcome of each round helps them adjust their choice of speed, potentially converging towards a stable pattern.
  • If \( x \) is near 100, drivers will likely converge to higher speeds over time. Conversely, if \( x \) is near 0, the tendency will be towards more cautious speeds.
The dynamic nature of repeated games illustrates how ongoing interactions and learning influence sustained strategic decision-making.

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Most popular questions from this chapter

Consider an asymmetric Cournot duopoly game, where the two firms have different costs of production. Firm 1 selects quantity \(q_{1}\) at a production cost of \(2 q_{1}\). Firm 2 selects quantity \(q_{2}\) and pays the production cost \(4 q_{2}\). The market price is given by \(p=12-q_{1}-q_{2}\). Thus, the payoff functions are \(u_{1}\left(q_{1}, q_{2}\right)=\left(12-q_{1}-q_{2}\right) q_{1}-2 q_{1}\) and \(u_{2}\left(q_{1}, q_{2}\right)=\left(12-q_{1}-q_{2}\right) q_{2}-4 q_{2}\). Calculate the firms' best-response functions \(B R_{1}\left(q_{2}\right)\) and \(B R_{2}\left(q_{1}\right)\), and find the Nash equilibrium of this game.

An island has two reefs that are suitable for fishing, and there are twenty fishers who simultaneously and independently choose at which of the two reefs ( 1 or 2 ) to fish. Each fisher can fish at only one reef. The total number of fish harvested at a single reef depends on the number of fishers who choose to fish there. The total catch is equally divided between the fishers at the reef. At reef 1 , the total harvest is given by \(f_{1}\left(r_{1}\right)=8 r_{1}-\frac{r_{1}^{2}}{2}\), where \(r_{1}\) is the number of fishers who select reef 1 . For reef 2 , the total catch is \(f_{2}\left(r_{2}\right)=4 r_{2}\), where \(r_{2}\) is the number of fishers who choose reef 2 . Assume that each fisher wants to maximize the number of fish that he or she catches. (a) Find the Nash equilibrium of this game. In equilibrium, what is the total number of fish caught? (b) The chief of the island asks his economics advisor whether this arrangement is efficient (i.e., whether the equilibrium allocation of fishers among reefs maximizes the number of fish caught). What is the answer to the chief's question? What is the efficient number of fishers at each reef? (c) The chief decides to require a fishing license for reef 1 , which would require each fisher who fishes there to pay the chief \(x\) fish. Find the Nash equilibrium of the resulting location-choice game between the fishers. Is there a value of \(x\) such that the equilibrium choices of the fishers results in an efficient outcome? If so, what is this value of \(x\) ?

Consider a strategic setting in which two geographically distinct firms (players 1 and 2) compete by setting prices. Suppose that consumers are uniformly distributed across the interval \([0,1]\), and each will buy either one unit or nothing. Firm 1 is located at 0 and firm 2 is located at 1 . Assume that the firms cannot change their locations; they can only select prices. Simultaneously and independently, firm 1 chooses a price \(p_{1}\) and firm 2 chooses a price \(p_{2}\). Suppose that the firms produce at zero cost and that due to a government regulation, they must set prices between 0 and 6 . As in the standard location game, consumers are sensitive to the distance they have to travel in order to purchase. But they are also sensitive to price. Consumers get a benefit of 6 from the good that is purchased, but they also pay a personal cost of \(c\) times the distance they have to travel to make the purchase. Assume that \(c\) is a positive constant. If the consumer at location \(x \in[0,1]\) purchases from firm 1, then this consumer's utility is \(6-c x-p_{1}\). If this consumer instead purchases from firm 2 , then her utility is \(6-c(1-x)-p_{2}\). If this consumer does not purchase the good, her utility is 0 . (a) Suppose that for given prices \(p_{1}\) and \(p_{2}\), every consumer purchases the good. That is, ignore the case in which prices are so high that some consumers prefer not to purchase. Find an expression for the "marginal consumer" who is indifferent between purchasing from firm 1 or firm \(2 .\) Denote this consumer's location as \(x^{*}\left(p_{1}, p_{2}\right)\). (b) Continue to assume that all consumers purchase at the prices you are analyzing. Note that firm 1's payoff (profit) is \(p_{1} x^{*}\left(p_{1}, p_{2}\right)\) and firm 2's payoff is \(p_{2}\left[1-x^{*}\left(p_{1}, p_{2}\right)\right]\). Calculate each firm's best response as a function of the other player's strategy. Also graph the best-response functions for the case of \(c=2\). (c) Find and report the Nash equilibrium of this game for the case in which \(c=2\). (d) As \(c\) converges to 0 , what happens to the firms' equilibrium profits? (e) What are the rationalizable strategies of this game for the case in which \(c=2\) ? (f) Find the Nash equilibrium of this game for the case in which \(c=8\).

Consider a more general Cournot model than the one presented in this chapter. Suppose there are \(n\) firms. The firms simultaneously and independently select quantities to bring to the market. Firm \(i\) 's quantity is denoted \(q_{i}\), which is constrained to be greater than or equal to zero. All of the units of the good are sold, but the prevailing market price depends on the total quantity in the industry, which is \(Q=\sum_{i=1}^{n} q_{i}\). Suppose the price is given by \(p=a-b Q\) and suppose each firm produces with marginal cost \(c\). There is no fixed cost for the firms. Assume \(a>c>0\) and \(b>0\). Note that firm \(i\) 's profit is given by \(u_{i}=p(Q) q_{i}-c q_{i}=(a-b Q) q_{i}-c q_{i}\). Defining \(Q_{-i}\) as the sum of the quantities produced by all firms except firm \(i\), we have \(u_{i}=\left(a-b q_{i}-b Q_{-i}\right) q_{i}-c q_{i}\). Each firm maximizes its own profit. (a) Represent this game in the normal form by describing the strategy spaces and payoff functions. (b) Find firm \(i\) 's best-response function as a function of \(Q_{-i}\). Graph this function. (c) Compute the Nash equilibrium of this game. Report the equilibrium quantities, price, and total output. (Hint: Summing the best-response functions over the different players will help.) What happens to the equilibrium price and the firm's profits as \(n\) becomes large? (d) Show that for the Cournot duopoly game \((n=2)\), the set of rationalizable strategies coincides with the Nash equilibrium.

In the years 2000 and 2001 , the bubble burst for many Internet and computer firms. As they closed shop, some of the firms had to liquidate sizable assets, such as inventories of products. Suppose eToys is going out of business and the company seeks a buyer for a truckload of Elmo dolls in its warehouse. Imagine that eToys holds an auction on eBay to sell the dolls and that two retailers (players 1 and 2 ) will bid for them. The rules of the auction are as follows: the retailers simultaneously and independently submit sealed bids and then eToys gives the merchandise to the highest bidder, who must pay his bid. It is common knowledge that the retailer who obtains the load of dolls can resell the load for a total of \(\$ 15,000\). Thus, if player \(i\) wins the auction with bid \(b_{i}\), then player \(i\) 's payoff is \(\$ 15,000-b_{i}\). The losing retailer gets a payoff of \(\$ 0\). If the retailers make the same bids \(\left(b_{1}=b_{2}\right)\), then eToys declares each player the winner with probability \(1 / 2\), in which case player \(i\) obtains an expected payoff of \((1 / 2)\left(15,000-b_{i}\right)\). What will be the winning bid in the Nash equilibrium of this auction game? If you can, describe the equilibrium strategies and briefly explain why this is an equilibrium. (Hint: This is similar to the Bertrand game.)
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