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Chapter 3: Problem 78

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 \(i^{\prime}\) 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 \(h\left(t_{j}\right)>\frac{1}{2}\).)

Short Answer

Expert verified
Both firms release their product at time \(T\). This is the Nash equilibrium.

Step by step solution

01

- Understand the Market Share Function

The function \(h(t)\) defines the market share captured by the first firm to release its product at time \(t\). This function increases from \(0\) at time \(0\) to \(1\) at time \(T\). Thus, \(0 \leq h(t) \leq 1\) and \(h(0)=0\), \(h(T)=1\).
02

- Define the Payoffs

Consider two firms, Firm A and Firm B. If Firm A releases its product at time \(t_A\) and Firm B releases its product at time \(t_B\): \If \(t_A < t_B\), Firm A captures \(h(t_A)\) of the market, and Firm B gets the remaining \(1 - h(t_A)\). \If \(t_B < t_A\), Firm B captures \(h(t_B)\) of the market, and Firm A gets the remaining \(1 - h(t_B)\). \If \(t_A = t_B\), both firms capture \(\frac{1}{2}\) of the market.
03

- Define the Best Response Function

The best response for a firm depends on the time \(t_j\) chosen by the competitor: \If \(h(t_j) < \frac{1}{2}\): the best response is to release the product slightly after \(t_j\) since the second firm will still get more than half the market, i.e., \({t_i = t_j + \epsilon}\). \If \(h(t_j) = \frac{1}{2}\): the best response is any time, as it will result in splitting the market equally no matter the choice. \If \(h(t_j) > \frac{1}{2}\): the best response is to release the product slightly before \(t_j\) to capture more than half the market, i.e., \({t_i = t_j - \epsilon}\).
04

- Determine the Nash Equilibrium

In a Nash equilibrium, neither firm can improve its payoff by unilaterally changing its release time. \Given the best response calculated previously, both firms release the product at time \(T\), as any firm releasing just before the other would not be better off due to the continuity and increase of \(h(t)\). At \(T\), releasing earlier does not reduce competition benefits, and thus, both firms are best off releasing at \(t = T\).

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Key Concepts

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

Strategic Game
In economic and business contexts, a strategic game refers to situations where players, often firms or individuals, make decisions that influence each other's outcomes. In the provided exercise, two firms are developing rival products and must decide on the optimal release time. This situation forms a strategic game as each firm's decision directly impacts the market share captured by both.
Each firm aims to maximize its market share by selecting a release time. The game's key elements are:
  • Players: The two firms releasing products.
  • Strategies: Each firm chooses a release time between 0 and T.
  • Payoffs: The share of the market each firm captures based on its chosen release time.
The interaction between these elements defines the essence of a strategic game. The firms must strategically choose release times while considering the potential choices of their competitor.
Market Share Function
The market share function, denoted as \(h(t)\), is pivotal in this scenario. It represents the portion of the market captured by the first firm to release its product at time \(t\).
Key characteristics of \(h(t)\):
  • It increases over time from 0 at time 0 to 1 at time T.
  • Initially, \(h(0)=0\), meaning no market share is captured at the very start.
  • When the time reaches T, the first firm captures the entire market: \(h(T)=1\).
This function is crucial for calculating payoffs:
  • If one firm releases its product at time \(t_A\) before the other firm's time \(t_B\), it captures \(h(t_A)\), and the second firm captures the remainder, \(1 - h(t_A)\).

Here鈥檚 a breakdown of the scenarios:
  • If \(t_A < t_B\), Firm A captures \(h(t_A)\).
  • If \(t_B < t_A\), Firm B captures \(h(t_B)\).
  • If both firms release simultaneously, each captures half the market: \(\frac{1}{2}\).
Best Response Function
To find an optimal strategy, each firm must consider what the rival will do. This is captured by the concept of the best response function, which defines the best action given the competitor's choice.
The best response function varies based on the competitor's chosen time \(t_j\):
  • If \(h(t_j) < \frac{1}{2}\): The best response is to release slightly after \(t_j\) (\(t_i = t_j + \epsilon\)), as the second firm will still obtain more than half of the market.
  • If \(h(t_j)=\frac{1}{2}\): The best response is any time, resulting in an equal split of the market share: \(\frac{1}{2}\) for each firm.
  • If \(h(t_j) > \frac{1}{2}\): The best response is to release slightly before \(t_j\) (\(t_i = t_j - \epsilon\)), capturing more than half the market.
The firms use this function to adjust their strategies and maximize their payoffs depending on the competitor's decision.
Nash Equilibrium
The Nash Equilibrium is a fundamental concept in game theory where no player can improve their payoff by unilaterally changing their strategy. In this exercise, the Nash equilibrium is found by considering each firm's best response.
  • If one firm were to release at time \(t_A\) just before the other firm's time \(t_B\), it wouldn't gain any advantage due to the increasing nature of \(h(t)\).
  • Since \(h(T)=1\), both firms delaying until time \(T\) maximizes their respective gains.

Thus, the Nash equilibrium in this game is for both firms to release their products at time \(T\). This outcome ensures neither firm can improve their market share by altering their release time unilaterally, making it the optimal strategy for both competitors.

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Most popular questions from this chapter

(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}

(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 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.)

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.)

(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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