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Chapter 3: Problem 24

(A fight) Each of two people has one unit of a resource. Each person chooses how much of the resource to use in fighting the other individual and how much to use productively. If each person \(i\) devotes \(y_{i}\) to fighting then the total output is \(f\left(y_{1}, y_{2}\right) \geq 0\) and person \(i\) obtains the fraction \(p_{i}\left(y_{1}, y_{2}\right)\) of the output, where $$ p_{i}\left(y_{1}, y_{2}\right)= \begin{cases}1 & \text { if } y_{i}>y_{j} \\\ \frac{1}{2} & \text { if } y_{i}=y_{j} \\ 0 & \text { if } y_{i}

Short Answer

Expert verified
The Nash equilibrium is \( y_1 = y_2 = 1 \), where both players devote all their resources to fighting.

Step by step solution

01

- Identify the Players

In this game, there are two players, Player 1 and Player 2.
02

- Identify the Strategies

Each player has a resource of 1 unit. Each player can choose to allocate some part (or all) of this resource to fighting, denoted as \( y_i \), where \( 0 \leq y_i \leq 1 \). Hence, the strategy sets for both players can be expressed as the interval [0, 1].
03

- Identify the Payoffs

The output function is given by \( f(y_1, y_2) = 2 - y_1 - y_2 \). The fraction of the output each player gets depends on their fighting resource allocation \( y_i \) relative to the other player's allocation \( y_j \). The payoffs are given by: \[ p_1(y_1, y_2) = \begin{cases} 1 & \text{if } y_1 > y_2 \ 0.5 & \text{if } y_1 = y_2 \ 0 & \text{if } y_1 < y_2 \end{cases} \]\[ p_2(y_1, y_2) = \begin{cases} 1 & \text{if } y_2 > y_1 \ 0.5 & \text{if } y_1 = y_2 \ 0 & \text{if } y_2 < y_1 \end{cases} \].Therefore, Player 1鈥檚 payoff is \( P_1 = p_1(y_1, y_2) \times f(y_1, y_2) \) and similarly, Player 2鈥檚 payoff is \( P_2 = p_2(y_1, y_2) \times f(y_1, y_2) \).
04

- Consider Condition \( y_1 eq y_2 \)

First, analyze cases where \( y_1 eq y_2 \). If \( y_1 > y_2 \), Player 1 will receive all the output, so Player 1's payoff \( P_1 = f(y_1, y_2) = 2 - y_1 - y_2 \). Player 2's payoff \( P_2 = 0 \). Conversely, if \( y_1 < y_2 \), Player 2 will receive all the output, so Player 2's payoff \( P_2 = f(y_1, y_2) = 2 - y_1 - y_2 \) and \( P_1 = 0 \). In this scenario, each player has an incentive to outbid the other by increasing their fighting resource allocation slightly, leading them both to keep increasing their fighting efforts.
05

- Consider Condition \( y_1 = y_2 < 1 \)

Next, analyze cases where \( y_1 = y_2 = y < 1 \). Both players receive equal shares, so their payoffs are: \[ P_1 = P_2 = 0.5 \times (2 - y - y) = 1 - y \].Since both players get half the total output, if either player deviates up slightly, they get the whole output still attainable, creating an incentive to increase fighting allocation.
06

- Consider Condition \( y_1 = y_2 = 1 \)

Consider when both players fully allocate their resources to fighting, \( y_1 = y_2 = 1 \). Here, \( f(1, 1) = 0 \) and both players鈥� payoffs are \( 0.5 \times 0 = 0 \). If this allocation is Nash, no one can gain unilaterally by deviating since deviation would imply reducing their fighting effort and obviously making him lose the whole reward. Thus exerting maximum resources for fighting here is Nash.

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Key Concepts

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

Nash equilibrium
Understanding the Nash equilibrium in this strategic game involves players choosing their best strategy considering the strategies of others. In this scenario, we first analyze the case when both players devote different amounts of resources to fighting. The player with greater resources allocated to fighting wins all the output. Thus, both players will have the incentive to increase their fighting resource allocation continually. Finally, an equilibrium where both players exactly choose all their resources for fighting and get zero payoffs reflects a stable Nash situation as any deviation leads to worse outcomes for both of them. In summary, a Nash equilibrium exists where both players' strategies maximize their respective payoffs, given the other player's strategy.
resource allocation
Resource allocation in this game refers to how each player decides to use their one unit of resource between fighting and productive activities. Here, players allocate their resources strategically to maximize their outcomes. The total output, represented by the function \( f(y_{1}, y_{2}) = 2 - y_{1} - y_{2} \), is negatively impacted by the amount of resources devoted to fighting by both players. Both players must balance allocating resources to fighting (which potentially increases their share of the output) and to productive uses (to maximize the overall output). Since the setup disincentivizes lower fighting resource allocation (through payoff matrices), strategic decisions naturally gravitate towards complete resource allocation to fighting if no coordination or cooperation mechanism exists.
payoff matrix
The payoff matrix helps visualize and calculate the respective payoffs for the players' chosen strategies. In this game, the payoffs depend on the fighting resource allocations and are calculated using the formula: \( P_1 = p_1(y_1, y_2) \times f(y_1, y_2) \) and \( P_2 = p_2(y_1, y_2) \times f(y_1, y_2 \). Simplifying, the descriptions yield \( p_i(y_1, y_2) \) as either 1 if the player allocates more resources to fighting, 0.5 if they allocate the same, and 0 otherwise. The payoff matrix then is essentially zero for weak strategies, half the decreased output for ties, and only fully positive in the case of winning greater output percentage by higher fighting effort. Visualization via the payoff matrix underscores the strategic advantage directly linked to aggressive fighting stances under given rules.
strategic decision making
Strategic decision making in this context involves players making choices based on the anticipated actions of the other player. Each player's goal is to maximize their payoff, leading them to decide the best level to allocate their resource to fighting. The crucial insights here involve recognizing that any small advantage in fighting results in the entire output for the winning player. Consequently, the strategic decision leads both players to invest all resources in fighting, given the others鈥� perceived rational responses, dictating that no other mixed or intermediate strategy steers better outcomes sustainably. The repeated game theory dynamic hints often rationalizes descent into resource-consuming conflicts naturally devoid of externally enforced cooperative frameworks.

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Most popular questions from this chapter

(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.

Consider the extent to which the analysis depends upon the demand function \(D\) taking the specific form \(D(p)=\alpha-p .\) Suppose that \(D\) is any function for which \(D(p) \geq 0\) for all \(p\) and there exists \(\bar{p}>c\) such that \(D(p)>0\) for all \(p \leq \bar{p} .\) Is \((c, c)\) still a Nash equilibrium? Is it still the only Nash equilibrium?

(Electoral competition with asymmetric voters' preferences) Consider a variant of Hotelling's model in which voters's preferences are asymmetric. Specifically, suppose that each voter cares twice as much about policy differences to the left of her favorite position than about policy differences to the right of her favorite position. How does this affect the Nash equilibrium? In the model considered so far, no candidate has the option of staying out of the race. Suppose that we give each candidate this option; assume that it is better than losing and worse than tying for first place. Then the Nash equilibrium remains as before: both players enter the race and choose the position \(m\). The direct argument differs from the one before only in that in addition we need to check that there is no equilibrium in which one or both of the candidates stays out of the race. If one candidate stays out then, given the other candidate's position, she can enter and either win outright or tie for first place. If both candidates stay out, then either candidate can enter and win outright. The next exercise asks you to consider the Nash equilibria of this variant of the model when there are three candidates.

Consider Cournot's game in the case of an arbitrary number \(n\) of firms; retain the assumptions that the in-verse demand function takes the form (54.2) and the cost function of each firm \(i\) is \(\mathrm{C}_{i}\left(q_{i}\right)=c q_{i}\) 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 \(q_{\prime}\) this is the price at which the output is sold.

(Timing product release) Two firms are developing competing products for a market of fixed size. The longer a firm spends on development, the better its product. But the first firm to release its product has an advantage: the customers it obtains will not subsequently switch to its rival. (Once a person starts using a product, the cost of switching to an alternative, even one significantly better, is too high to make a switch worthwhile.) A firm that releases its product first, at time \(t\), captures the share \(h(t)\) of the market, where \(h\) is a function that increases from time 0 to time \(T\), with \(h(0)=0\) and \(h(T)=1 .\) The remaining market share is left for the other firm. If the firms release their products at the same time, each obtains half of the market. Each firm wishes to obtain the highest possible market share. Model this situation as a strategic game and find its Nash equilibrium (equilibria?). (When finding firm \(i\) 's best response to firm \(j\) 's release time \(t_{j}\), there are three cases: that in which \(h\left(t_{j}\right)<\frac{1}{2}\) (firm \(j\) gets less than half of the market if it is the first to release its product), that in which \(h\left(t_{j}\right)=\frac{1}{2}\), and that in which \(\left.h\left(t_{j}\right)>\frac{1}{2} .\right)\)
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