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Chapter 3: Problem 75

In the variant of Hotelling's model that captures competing firms' choices of product characteristics, show that when there are two firms the unique Nash equilibrium is \((m, m)\) (both firms offer the consumers' median favorite product) and when there are three firms there is no Nash equilibrium. (Start by arguing that when there are two firms whose products differ, either firm is better off making its product more similar to that of its rival.)

Short Answer

Expert verified
The Nash equilibrium with two firms is at (m, m). There is no Nash equilibrium with three firms due to continuous adjustment and instability.

Step by step solution

01

Understand Hotelling's Model

Hotelling's model is about product differentiation and location choice among firms. In this exercise, two firms must choose product characteristics to maximize their share of consumers.
02

Equilibrium with Two Firms

Suppose the two firms choose product characteristics such that their products are located at points, say, A and B on a line representing consumer preferences. Consumers will buy from the firm whose product characteristic is closest to their preference.
03

Profit Maximization for Two Firms

Consider that if firm A's product is located at point A and firm B's product is at point B, and A < B, each firm attracts consumers to its closest point. Firm A can then increase its market share by moving its product characteristic closer to point B, because this reduces the distance consumers who originally preferred B must now travel to reach A.
04

Both Firms Tend Toward the Median

Consequently, both firms will continuously adjust their product characteristics to be closer to each other. The equilibrium point is reached when both firms choose the median product characteristic (m) to capture an equal share of the market, as any deviation would reduce their respective consumers.
05

Nash Equilibrium for Two Firms

Thus, when there are two firms, the unique Nash equilibrium is when both firms choose the median product characteristic (m, m), since neither firm gains by altering its position.
06

Three Firms Analysis

In the case of three firms, suppose firms A, B, and C choose different positions on the line. Each firm would continually adjust their position to try and capture a larger market share by becoming more similar to the most popular positions. However, this continuous adjustment among three points with different positions leads to an unstable relationship where no single position can be chosen that benefits all simultaneously.
07

No Nash Equilibrium for Three Firms

Due to the instability caused by perpetual adjustment and no single stable position benefiting all firms simultaneously, there is no Nash equilibrium when there are three firms.

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Key Concepts

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

Nash equilibrium
In game theory, Nash equilibrium is a situation where no player can benefit by changing their strategy while the other players keep theirs unchanged.
In Hotelling's model, we can see this when two firms end up choosing the same product characteristic at the median preference point (m).
This becomes the Nash equilibrium because neither firm can improve their outcome by deviating from this strategy.
In simpler terms:
  • If Firm A and Firm B both choose the median product, moving away to another point will reduce their immediate consumer base.
  • This means they have no incentive to change, ensuring stability in their choices.
Hence, the Nash equilibrium for two firms is \((m, m)\) because it ensures both firms maximize their market share without benefiting from a new strategy.
Product differentiation
Product differentiation is when firms make products that are different, to attract different types of consumers.
In Hotelling's model, firms decide on product characteristics to attract consumers with varied preferences.
Firms aim to position their products strategically so they appeal to the largest group. Initially:
  • Two firms choose different points on the preference line (e.g., points A and B)
  • Consumers will buy from the firm whose product characteristic closest matches their preferences.
However, as businesses realize they can gain more consumers by becoming more similar (moving closer to each other), they adjust their characteristics.
By continuing this adjustment, both firms eventually offer very similar products centered around the median preference.
So, product differentiation helps firms find a balance between unique features and broad appeal.
Market share maximization
Maximizing market share is the main objective for firms in Hotelling's model.
Market share refers to the portion of consumers each firm successfully captures.
Firms continuously adjust their product characteristics to attract as many consumers as possible:
  • If Firm A's product is at point A and Firm B's product is at point B, each firm attracts consumers closest to their point.
  • If Firm A moves its product closer to point B, it appeals to more consumers from Firm B.
This constant adjustment leads both firms to pick the median product characteristic (m).
At this midpoint, each firm captures half of the market, because they are equidistant to all consumers. So, both firms maximize their market share by settling on this strategy.
Consumer preferences
Consumer preferences dictate how firms in Hotelling's model choose their product characteristics.
Preferences vary and can be mapped along a line, let's say from point A to point B.
Each consumer will purchase from the firm whose product is closest to their own preference:
  • Consumers at the far left will prefer a product closest to point A.
  • Consumers at the far right will prefer a product closest to point B.
Firms aim to attract as many consumers as possible, so understanding these preferences is crucial.
This is why two firms end up selecting the median product characteristics (m), as it sits in the middle of all consumer preferences.
By doing so, they ensure the maximum reach and appeal.
Two-agent models
Hotelling's model often utilizes two-agent models, where only two firms compete.
This simplifies the analysis and helps show how strategic positioning affects each one's market share.
In this model:
  • Each firm selects its strategy (product characteristic).
  • They observe the other's choice and adjust to gain more consumers.
For two-agent models, equilibrium is reached when firms pick product characteristics that neither finds beneficial to change (median in this case).
However, when extended to three firms, such equilibrium doesn't exist. Each firm continues to adjust, seeking an advantage, resulting in instability.
Thus, the two-agent model clearly demonstrates how firms interact and reach equilibrium in competitive scenarios.

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Most popular questions from this chapter

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

(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.

EXERCISE \(85.1\) (First-price sealed-bid auction) Show that in a Nash equilibrium of a first-price sealed-bid auction the two highest bids are the same, one of these bids is submitted by player 1, and the highest bid is at least \(v_{2}\) and at most \(v_{1}\). Show also that any action profile satisfying these conditions is a Nash equilibrium. In any equilibrium in which the winning bid exceeds \(v_{2}\), at least one player's bid exceeds her valuation. As in a second-price sealed-bid auction, such a bid seems "risky", because it would yield the bidder a negative payoff if it were to win. In the equilibrium there is no risk, because the bid does not win; but, as before, the fact that the bid has this property reduces the plausibility of the equilibrium. As in a second-price sealed-bid auction, the potential "riskiness" to player \(i\) of a bid \(b_{i}>v_{i}\) is reflected in the fact that it is weakly dominated by the bid \(v_{i}\), as shown by the following argument. \- If the other players' bids are such that player \(i\) loses when she bids \(b_{i}\), then the outcome is the same whether she bids \(b_{i}\) or \(v_{i}\). \- If the other players' bids are such that player \(i\) wins when she bids \(b_{i}\), then her payoff is negative when she bids \(b_{i}\) and zero when she bids \(v_{i}\) (whether or not this bid wins). However, in a first-price auction, unlike a second-price auction, a bid \(b_{i}b_{i}\) because if the other players' highest bid is less than \(b_{i}\) then both \(b_{i}\) and \(b_{i}^{\prime}\) win and \(b_{i}\) yields a lower price. Further, even though the bid \(v_{i}\) weakly dominates higher bids, this bid is itself weakly dominated, by a lower bid! If player \(i\) bids \(v_{i}\) her payoff is 0 regardless of the other players' bids, whereas if she bids less than \(v_{i}\) her payoff is either 0 (if she loses) or positive (if she wins). In summary, in a first-price sealed-bid auction (with perfect information), a player's bid of at least her valuation is weakly dominated, and a bid of less than her valuation is not weakly dominated. An implication of this result is that in every Nash equilibrium of a first- price sealed-bid auction at least one player's action is weakly dominated. However, this property of the equilibria depends on the assumption that a bid may be any number. In the variant of the game in which bids and valuations are restricted to be multiples of some discrete monetary unit \(\epsilon\) (e.g. a cent), an action profile \(\left(v_{2}-\epsilon, v_{2}-\epsilon, b_{3}, \ldots, b_{n}\right)\) for any \(b_{j} \leq v_{j}-\epsilon\) for \(j=3, \ldots, n\) is a Nash equilibrium in which no player's bid is weakly dominated. Further, every equilibrium in which no player's bid is weakly dominated takes this form. When \(\epsilon\) is small, each such equilibrium is close to an equilibrium \(\left(v_{2}, v_{2}, b_{3}, \ldots, b_{n}\right)\) (with \(b_{j} \leq v_{j}\) for \(j=3, \ldots, n)\) of the game with unrestricted bids. On this (somewhat \(a d\) hoc) basis, I select action profiles \(\left(v_{2}, v_{2}, b_{3}, \ldots, b_{n}\right)\) with \(b_{j} \leq v_{j}\) for \(j=3, \ldots, n\) as "distinguished" equilibria of a first-price sealed-bid auction. One conclusion of this analysis is that while both second-price and first- price auctions have many Nash equilibria, yielding a variety of outcomes, their distinguished equilibria yield the same outcome. (Recall that the distinguished equilibrium of a second-price sealed-bid auction is the action profile in which every player bids her valuation.) In every distinguished equilibrium of each game, the object is sold to player 1 at the price \(v_{2} .\) In particular, the auctioneer's revenue is the same in both cases. Thus if we restrict attention to the distinguished equilibria, the two auction forms are "revenue equivalent". The rules are different, but the players' equilibrium bids adjust to the difference and lead to the same outcome: the single Nash equilibrium in which no player's bid is weakly dominated in a second-price auction yields the same outcome as the distinguished equilibria of a first-price auction.

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.

(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.
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