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Chapter 3: Problem 77

Consider the variant of the War of Attrition in which each player attaches no value to the time spent waiting for the other player to concede, but the object in dispute loses value as time passes. (Think of a rotting animal carcass or a melting ice cream cone.) Assume that the value of the object to each player \(i\) after \(t\) units of time is \(v_{i}-t\) (and the value of a \(50 \%\) chance of obtaining the object is \(\left.\frac{1}{2}\left(v_{i}-t\right)\right) .\) Specify the strategic game that models this sit- uation (take care with the payoff functions). Construct the analogue of Figure \(76.1\), find the players' best response functions, and hence find the Nash equilibria of the game. The War of Attrition is an example of a "game of timing", in which each player's action is a number and each player's payoff depends sensitively on whether her action is greater or less than the other player's action. In many such games, each player's strategic variable is the time at which to act, hence the name "game of timing". The next two exercises are further examples of such games. (In the first the strategic variable is time, whereas in the second it is not.)

Short Answer

Expert verified
Players choose waiting times that equalize the value of the object being contested, resulting in a Nash equilibrium.

Step by step solution

01

Define the players and the strategies

In this game, there are two players. Each player's strategy is a choice of how long to wait before conceding. We can denote the waiting time of Player 1 as \(t_1\) and the waiting time of Player 2 as \(t_2\). The values of the object to Player 1 and Player 2 are \(v_1\) and \(v_2\) respectively.
02

Define the payoff functions

If Player 1 waits for a time \(t_1\) and Player 2 waits for a time \(t_2\), the following cases determine their payoffs: 1. If \(t_1 < t_2\), Player 1 concedes first, and their payoff is \(0\). Player 2 obtains the object and their payoff is \(v_2 - t_2\). 2. If \(t_1 > t_2\), Player 2 concedes first, and their payoff is \(0\). Player 1 obtains the object and their payoff is \(v_1 - t_1\).3. If \(t_1 = t_2\), both players have an equal chance of winning the object. Player 1's expected payoff is \(\frac{1}{2}(v_1 - t_1)\) and Player 2's expected payoff is \(\frac{1}{2}(v_2 - t_2)\).
03

Analyze the best response functions

Player 1's best response to Player 2's strategy \(t_2\) is to choose \(t_1\) such that their expected payoff is maximized. Similarly, Player 2's best response to Player 1's strategy \(t_1\) is to choose \(t_2\) to maximize their expected payoff. This involves setting up and solving the payoff functions described previously.
04

Find the Nash equilibria

To find the Nash equilibria, we look for a pair \((t_1, t_2)\) where each player's chosen waiting time is the best response to the other player's waiting time. By analyzing when both players have no incentive to deviate, we determine the equilibrium conditions. For this particular setup, the equilibrium is when both players choose the same waiting time \(t_i\) such that they are indifferent between waiting and conceding.

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Key Concepts

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

War of Attrition
The War of Attrition is a strategic game often known as a 'game of timing'. In this game, players compete by enduring a cost over time to win a valuable object. In our exercise, the value of the object decreases over time, making it essential to decide wisely when to stop waiting. Imagine competing over a rotting carcass. If you wait too long, its value diminishes. Each player must weigh their willingness to endure the cost against their competitor鈥檚 patience.
Strategic Game
A strategic game involves players making choices that affect their outcomes based on the choices of others. In the War of Attrition, each player picks a time to concede. The goal is to outlast your opponent but also be strategic about not over-waiting since the prize value decreases over time.
Nash Equilibrium
A Nash equilibrium is a set of strategies where no player can benefit by unilaterally changing their strategy. For our game, it's where both players choose waiting times that best respond to each other. Here, if both players choose the same time, they reach equilibrium because neither has an incentive to change their waiting strategy.
Payoff Functions
In this game, players' payoffs depend on both the strategies they choose and the timing of those choices. For instance, if Player 1 waits longer than Player 2 and wins, their payoff is the object's value minus their waiting time. The payoff functions describe how each player鈥檚 outcome changes based on both their and their opponent鈥檚 waiting times.

If Player 1 waits less time than Player 2 (i.e., they concede first), they gain nothing, and Player 2 gets the prize value minus their waiting cost.

If both players wait the same amount of time, they split the prize value half-and-half, each considering the time they waited.

The goal is to find the strategy combination where both players maximize their expected outcomes given the other鈥檚 strategy. This is done by determining their best responses and seeking the Nash equilibrium.

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Most popular questions from this chapter

(Citizen-candidates) Consider a game in which the players are the citizens. Any citizen may, at some cost \(c>0\), become a candidate. Assume that the only position a citizen can espouse is her favorite position, so that a citizen's only decision is whether to stand as a candidate. After all citizens have (simultaneously) decided whether to become candidates, each citizen votes for her favorite candidate, as in Hotelling's model. Citizens care about the position of the winning candidate; a citizen whose favorite position is \(x\) loses \(\left|x-x^{*}\right|\) if the winning candidate's position is \(x^{*}\). (For any number \(z,|z|\) denotes the absolute value of \(z:|z|=z\) if \(z>0\) and \(|z|=-z\) if \(z<0 .\) ) Winning confers the benefit \(b\). Thus a citizen who becomes a candidate and ties with \(k-1\) other candidates for first place obtains the payoff \(b / k-c\); a citizen with favorite position \(x\) who becomes a candidate and is not one of the candidates tied for first place obtains the payoff \(-\left|x-x^{*}\right|-c\), where \(x^{*}\) is the winner's position; and a citizen with favorite position \(x\) who does not become a candidate obtains the payoff \(-\left|x-x^{*}\right|\), where \(x^{*}\) is the winner's position. Assume that for every position \(x\) there is a citizen for whom \(x\) is the favorite position. Show that if \(b \leq 2 c\) then the game has a Nash equilibrium in which one citizen becomes a candidate. Is there an equilibrium (for any values of \(b\) and \(c\) ) in which two citizens, each with favorite position \(m\), become candidates? Is there an equilibrium in which two citizens with favorite positions different from \(m\) become candidates? Hotelling's model assumes a basic agreement among the voters about the ordering of the positions. For example, if one voter prefers \(x\) to \(y\) to \(z\) and another voter prefers \(y\) to \(z\) to \(x\), no voter prefers \(z\) to \(x\) to \(y\). The next exercise asks you to study a model that does not so restrict the voters' preferences.

(Nash equilibrium of second-price sealed-bid auction) Find a Nash equilibrium of a second-price sealed-bid auction in which player \(n\) obtains the object. Player 2 's bid in this equilibrium exceeds her valuation, and thus may seem a little rash: if player 1 were to increase her bid to any value less than \(v_{1}\), player 2 's payoff would be negative (she would obtain the object at a price greater than her valuation). This property of the action profile does not affect its status as an equilibrium, because in a Nash equilibrium a player does not consider the "risk" that another player will take an action different from her equilibrium action; each player simply chooses an action that is optimal, given the other players' actions. But the property does suggest that the equilibrium is less plausible as the outcome of the auction than the equilibrium in which every player bids her valuation. The same point takes a different form when we interpret the strategic game as a model of events that unfold over time. Under this interpretation, player 2's action \(v_{1}\) means that she will continue bidding until the price reaches \(v_{1}\). If player 1 is sure that player 2 will continue bidding until the price is \(v_{1}\), then player 1 rationally stops bidding when the price reaches \(v_{2}\) (or, indeed, when it reaches any other level at most equal to \(v_{1}\) ). But there is little reason for player 1 to believe that player 2 will in fact stay in the bidding if the price exceeds \(v_{2}\) : player 2 's action is not credible, because if the bidding were to go above \(v_{2}\), player 2 would rationally withdraw. The weakness of the equilibrium is reflected in the fact that player 2 's bid \(v_{1}\) is weakly dominated by the bid \(v_{2} .\) More generally, in a second-price sealed-bid auction (with perfect information), a player's bid equal to her valuation weakly dominates all her other bids. That is, for any bid \(b_{i} \neq v_{i}\), player \(i^{\prime}\) s bid \(v_{i}\) is at least as good as \(b_{i}\), no matter what the other players bid, and is better than \(b_{i}\) for some actions of the other players. (See Definition 45.1.) A player who bids less than her valuation stands not to win in some cases in which she could profit by winning (when the highest of the other bids is between her bid and her valuation), and never stands to gain relative to the situation in which she bids her valuation; a player who bids more than her valuation stands to win in some cases in which she obtains a negative payoff by doing so (when the highest of the remaining bids is between her valuation and her bid), and never stands to gain relative to the situation in which she bids her valuation. The key point is that in a second-price auction, a player who changes her bid does not lower the price she pays, but only possibly changes her status from that of a winner into that of a loser, or vice versa. A precise argument is shown in Figure 84.1, which compares player \(i^{\prime}\) s payoffs to the bid \(v_{i}\) with her payoffs to a bid \(b_{i}

(Cournot's game with many firms) Consider Cournot's game in the case of an arbitrary number \(n\) of firms; retain the assumptions that the inverse demand function takes the form (54.2) and the cost function of each firm \(i\) is \(C_{i}\left(q_{i}\right)=c q_{i}\) for all \(q_{i}\), with \(c<\alpha .\) Find the best response function of each firm and set up the conditions for \(\left(q_{1}^{*}, \ldots, q_{n}^{*}\right)\) to be a Nash equilibrium (see \(\left.(34.3)\right)\), assuming that there is a Nash equilibrium in which all firms' outputs are positive. Solve these equations to find the Nash equilibrium. (For \(n=2\) your answer should be \(\left(\frac{1}{3}(\alpha-c), \frac{1}{3}(\alpha-c)\right)\), the equilibrium found in the previous section. First show that in an equilibrium all firms produce the same output, then solve for that output. If you cannot show that all firms produce the same output, simply assume that they do.) Find the price at which output is sold in a Nash equilibrium and show that this price decreases as \(n\) increases, approaching \(c\) as the number of firms increases without bound. The main idea behind this result does not depend on the assumptions on the inverse demand function and the firms' cost functions. Suppose, more generally, that the inverse demand function is any decreasing function, that each firm's cost function is the same, denoted by \(C\), and that there is a single output, say \(q\), at which the average cost of production \(C(q) / q\) is minimal. In this case, any given total output is produced most efficiently by each firm's producing \(q\), and the lowest price compatible with the firms' not making losses is the minimal value of the average cost. The next exercise asks you to show that in a Nash equilibrium of Cournot's game in which the firms' total output is large relative to \(\underline{q}\), this is the price at which the output is sold.

(Third-price auction) Consider a third-price sealed-bid auction, which differs from a first- and a second-price auction only in that the winner (the person who submits the highest bid) pays the third highest price. (Assume that there are at least three bidders.) \(a\). Show that for any player \(i\) the bid of \(v_{i}\) weakly dominates any lower bid, but does not weakly dominate any higher bid. (To show the latter, for any bid \(b_{i}>v_{i}\) find bids for the other players such that player \(i\) is better off bidding \(b_{i}\) than bidding \(v_{i}\).) b. Show that the action profile in which each player bids her valuation is not a Nash equilibrium. c. Find a Nash equilibrium. (There are ones in which every player submits the same bid.) 3.5.4 Variants Uncertain valuations One respect in which the models in this section depart from reality is in the assumption that each bidder is certain of both her own valuation and every other bidder's valuation. In most, if not all, actual auctions, information is surely less perfect. The case in which the players are uncertain about each other's valuations has been thoroughly explored, and is discussed in Section 9.7. The result that a player's bidding her valuation weakly dominates all her other actions in a second-price auction survives when players are uncertain about each other's valuations, as does the revenue- equivalence of first- and second-price auctions under some conditions on the players' preferences. Common valuations In some auctions the main difference between the bidders is not that the value the object differently but that they have different information about its value. For example, the bidders for an oil tract may put similar values on any given amount of oil, but have different information about how much oil is in the tract. Such auctions involve informational considerations that do not arise in the model we have studied in this section; they are studied in Section 9.7.3. Multi-unit auctions In some auctions, like those for Treasury Bills (short- term) government bonds) in the USA, many units of an object are available, and each bidder may value positively more than one unit. In each of the types of auction described below, each bidder submits a bid for each unit of the good. That is, an action is a list of bids \(\left(b^{1}, \ldots, b^{k}\right)\), where \(b^{1}\) is the player's bid for the first unit of the good, \(b^{2}\) is her bid for the second unit, and so on. The player who submits the highest bid for any given unit obtains that unit. The auctions differ in the prices paid by the winners. (The first type of auction generalizes a first-price auction, whereas the next two generalize a second-price auction.) Discriminatory auction The price paid for each unit is the winning bid for that unit. Uniform-price auction The price paid for each unit is the same, equal to the highest rejected bid among all the bids for all units. Vickrey auction A bidder who wins \(k\) objects pays the sum of the \(k\) highest rejected bids submitted by the other bidders. The next exercise asks you to study these auctions when two units of an object are available.

(Electoral competition with three candidates) Consider a variant of Hotelling's model in which there are three candidates and each candidate has the option of staying out of the race, which she regards as better than losing and worse than tying for first place. Use the following arguments to show that the game has no Nash equilibrium. First, show that there is no Nash equilibrium in which a single candidate enters the race. Second, show that in any Nash equilibrium in which more than one candidate enters, all candidates that enter tie for first place. Third, show that there is no Nash equilibrium in which two candidates enter the race. Fourth, show that there is no Nash equilibrium in which all three candidates enter the race and choose the same position. Finally, show that there is no Nash equilibrium in which all three candidates enter the race, and do not all choose the same position.
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