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Chapter 3: Problem 23

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

Short Answer

Expert verified
The Nash Equilibrium occurs when both firms release their products at the same time, at h(t) = 0.5.

Step by step solution

01

Understand the Problem

Two firms are competing to release a product to a market with fixed size. The market share depends on how long they spend on development. If Firm A releases its product first at time t, it captures share h(t) of the market where h(t) increases from 0 to 1 between times 0 and T. If both firms release at the same time, they share the market equally.
02

Define the Firms' Objectives

Both firms aim to maximize their market share. Firm i wants to choose a release time t_i considering the release time t_j of Firm j to capture the highest possible market share. We have three scenarios to consider based on the value of h(t_j).
03

Case 1: Firm j releases early (h(t_j) < 0.5)

If Firm j releases its product early such that h(t_j) < 0.5, Firm i can release its product slightly later to capture the remaining market share. In this case, Firm i鈥檚 best response is to choose a time t_i such that t_i > t_j but close enough to make significant market share gains.
04

Case 2: Firm j releases at midpoint (h(t_j) = 0.5)

If Firm j releases its product when h(t_j) = 0.5, this indicates an even split is possible if Firm i releases its product at the same time. Firm i should also release at the same time to capture an equal market share. The best response for Firm i is t_i = t_j.
05

Case 3: Firm j releases late (h(t_j) > 0.5)

If Firm j releases its product late such that h(t_j) > 0.5, Firm i can release its product earlier to capture the market before Firm j. Firm i's best response is to release at time t_i < t_j to capture part of the market while ensuring h(t_i) >= (1 - h(t_j)).
06

Identify Nash Equilibrium

A Nash Equilibrium exists when neither firm can improve their market share by unilaterally changing their release time. Given the three cases and mutual competition: both firms are incentivized to release at the midpoint when h(t) = 0.5, ensuring an equal share because any deviation would lead to a imbalance or reduced share.

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Key Concepts

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

Nash Equilibrium
The Nash Equilibrium is a key concept in game theory describing a stable state where no player can benefit by changing their strategy while the other players keep theirs unchanged. For our case, it means: when both firms release their products at the same time, each capturing half the market. If either tries to change the release time unilaterally, they risk getting a smaller market share.

This equilibrium happens because each firm understands that rushing to release first might lead their product to be inferior due to less development time. Similarly, waiting too long means the rival firm captures a significant market share first. Therefore, releasing at the midpoint where both firms agree guarantees an equal share, making it a mutual best response.
Strategic Game
A strategic game in economics involves players making decisions to maximize their payoffs considering the strategies of others. In the product release timing problem, the two firms are the players and their strategies are the release times.

The objective: maximize market share. This situation creates a strategic interaction where decisions depend on the anticipated actions of the competitor. Each firm needs to carefully consider when their rival plans to release, adjusting their timing accordingly. This interplay of strategies leads to the critical formation of responses ensuring the best possible outcome for each firm.
Market Competition
Market competition refers to the rivalry between firms seeking to capture consumer interest and maximize market share. Here, two firms are developing competing products for a fixed market.

The firm's strategies include choosing the precise time for product release balancing development quality and market capture. The first mover gets consumers early but risk releasing an inferior product. The second mover can develop a superior product but risks losing early adopters to a competitor. Thus, fierce competition shapes the strategic decisions in timing the product release.
Product Release Timing
Timing in product release is crucial in competitive markets. Releasing too early or late can dramatically affect market share.

In the example, releasing early captures initial market but a less developed product. Waiting ensures a better product, but risks the competitor securing the market first. Hence, the strategic balance becomes key, typically leading both firms aiming for an optimal midpoint to maximize potential market share while maintaining product quality. Each firm must analyze the market dynamics and competitor behavior to decide on the optimal release time, thus ensuring a strategic advantage.

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Most popular questions from this chapter

(Electoral competition for more general preferences) There is a finite number of positions and a finite, odd, number of voters. For any positions \(x\) ind \(y\), each voter either prefers \(x\) to \(y\) or prefers \(y\) to \(x\). (No voter regards any two positions as equally desirable.) We say that a position \(x^{*}\) is a Condorcet winner if for very position \(y\) different from \(x^{*}\), a majority of voters prefer \(x^{*}\) to \(y\). \(a\). Show that for any configuration of preferences there is at most one Condorcet winner. b. Give an example in which no Condorcet winner exists. (Suppose there are three positions \((x, y\), and \(z)\) and three voters. Assume that voter 1 prefers \(x\) to \(y\) to \(z\). Construct preferences for the other two voters such that one voter prefers \(x\) to \(y\) and the other prefers \(y\) to \(x\), one prefers \(x\) to \(z\) and the other prefers \(z\) to \(x\), and one prefers \(y\) to \(z\) and the other prefers \(z\) to \(y\). The preferences you construct must, of course, satisfy the condition that a voter who prefers \(a\) to \(b\) and \(b\) to \(c\) also prefers \(a\) to \(c\), where \(a, b\), and \(c\) are any positions.) c. Consider the strategic game in which two candidates simultaneously choose positions, as in Hotelling's model. If the candidates choose different positions, each voter endorses the candidate whose position she prefers, and the candidate who receives the most votes wins. If the candidates choose the same position, they tie. Show that this game has a unique Nash equilibrium if the voters' preferences are such that there is a Condorcet winner, and has no Nash equilibrium if the voters' preferences are such that there is no Condorcet winner. A variant of Hotelling's model of electoral competition can be used to analyze he choices of product characteristics by competing firms in situations in which lext exercise you are asked to show that the Nash equilibria of this game in the fase of two or three firms are the same as those in Hotelling's model of electoral competition.

(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.

(Multi-unit auctions) Two units of an object are available. There are \(n\) bidders. Bidder \(i\) values the first unit that she obtains at \(v_{i}\) and the second unit at \(w_{i}\), where \(v_{i}>w_{i}>0\). Each bidder submits two bids; the two highest bids win. Retain the tie-breaking rule in the text. Show that in discriminatory and uniform-price auctions, player \(i^{\prime}\) s action of bidding \(v_{i}\) and \(w_{i}\) does not dominate all her other actions, whereas in a Vickrey auction it does. (In the case of a Vickrey auction, consider separately the cases in which the other players' bids are such that player \(i\) wins no units, one unit, and two units when her bids are \(v_{i}\) and \(w_{i}\).) Goods for which the demand exceeds the supply at the going price are sometimes sold to the people who are willing to wait longest in line. We can model such situations as multi-unit auctions in which each person's bid is the amount of time she is willing to wait.

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?

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.
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