Rent shrinking.

I. Introduction

Economic models of such diverse activities as rent seeking, patent races, wars, and election campaign spending often employ the assumption that the competitors are chasing a prize of a fixed size.(1) In many circumstances, however, the size of the prize that goes to the winner is a decreasing function of the competitive expenditures of the losers. In a patent race, for example, the value of the patent to the winning firm will often be lower if other firms have been actively competing. The research expenditures of the close substitutes provided by the losing firms makes the patent less valuable to the winning firm. Anticipating this behavior, firms may have less incentive to engage in research and development when the actions of their competitors lower the value of the patent. A second example involves the market for corporate control. A takeover competition can lower the value of the firm that is the prize if the firm's current management responds to the takeover threat with a poison pill. The value of the firm may be lowered even if the takeover attempt is unsuccessful, i.e., if the current management wins the contest.

This article examines situations where the size of a prize to the winner falls as the expenditures of competing firms increase, a phenomenon we call "rent shrinking." A commonly used model of rent seeking is modified to incorporate rent shrinking. Rent shrinking lowers the rent seeking expenditures of the competing firms.(2) Collusive behavior, which acts to reduce the number of competitors, is more likely to be profitable with rent shrinking. By reducing the number of competitors, collusion increases the prize available to the colluding firms, ceteris paribus. This represents an additional benefit to collusion available in a rent shrinking environment.

In some circumstances, the shrinking of the rent to the winning firm that arises from opponents' expenditures is mirrored by the consideration that the losing firms' expenditures were not pure losses. For example, losers in a patent race may still be positioned to capture some rent, if they can successfully innovate around the patent. Such effects can be incorporated into the rent shrinking model, by assuming that losing firms are reimbursed for some percentage of their expenditures. This approach also serves to unify the standard rent seeking game with the rent shrinking game. If none of the expenditures of losing firms are reimbursed, the game is one of pure rent shrinking. If the entire expenditures of losers are reimbursed, the standard rent seeking game emerges. As the share of losers' expenditures that are returned goes from 0 to 1, the game moves from pure rent shrinking to one of pure rent seeking, and an individual firm's expenditures monotonically increase.

The basic rent seeking and rent shrinking models are presented in section II, where it is shown that rent shrinking reduces the expenditures of firms relative to rent seeking. Section III examines the potential for collusive behavior by firms. Section IV includes the possibility that a fraction of the expenditures of losing firms may be returned, an approach that generalizes the rent seeking and rent shrinking games; collusive behavior is also examined in this more general framework. Section V presents conclusions.

II. The Model

The standard rent seeking game consists of n risk neutral firms competing for a fixed rent of size [Pi].(3) Let [x.sub.i] [greater than or equal to] 0 represent the rent seeking expenditures of firm i, i = l, . . ., n. The probability that firm i wins the prize, [p.sub.i], is given by firm i's share of total rent seeking expenditures: [p.sub.i] = [x.sub.i]/([x.sub.i] + [summation of][x.sub.j]where j[not equal to]i). Firm i chooses expenditures [x.sub.i] to maximize its expected utility [U.sub.i] = ([p.sub.i] [center dot] [Pi]) - [x.sub.i]. At the symmetric Nash equilibrium, firm i's rent seeking expenditures are [Mathematical Expression Omitted], and its expected utility is [Mathematical Expression Omitted]. Total rent seeking expenditures are (n - 1)[Pi]/n. As the number of competing firms n gets large, total rent seeking expenditures approach the size of the rent [Pi], though an individual firm's rent seeking expenditures fall as n rises.(4)

The standard rent seeking game can be modified to incorporate rent shrinking. Let the original prize remain fixed at size [Pi]. If firm i, i = 1, . . ., n, is the winner, however, firm i only gets to enjoy a prize of size [Pi][prime] = [Pi] - [summation of][x.sub.j] where j[not equal to]i. The prize that the winning firm receives is lowered by the combined expenditures of the other firms.(5)

The problem for firm i is to maximize expected profits [U.sub.i] = ([p.sub.i] [multiplied by] [Pi][prime]) - [x.sub.i]. Solving from the first-order conditions, the symmetric Nash equilibrium expenditures for firm i in the rent shrinking game are [x[prime].sub.i] = (n - 1)[Pi]/(2[n.sup.2] - 2n + 1). For n [greater than] 1, [Mathematical Expression Omitted]: a firm's expenditures in a rent shrinking contest are less than a firm's expenditures in a rent seeking game, holding the number of firms constant. An individual firm's expenditures in a rent shrinking game fall as the number of firms increases, as in the rent seeking case. Firm i's expected utility in the rent shrinking game is [U[prime].sub.i] = [Pi]/(2[n.sup.2] - 2n + 1), which is below its expected utility in the standard rent seeking game for n [greater than] 1. Total rent shrinking expenditures are n [center dot] [x[prime].sub.i] = n(n - 1)[Pi]/(2[n.sup.2] - 2n + 1). A central result in the rent seeking model is that total rent seeking expenditures approach the size of the rent as the number of competitors gets large, i.e., the rent is totally dissipated as n [approaches] [infinity] [9; 13]. In the rent shrinking game, as n gets large, total expenditures approach one half of the original rent, [Pi]. The amount [Pi]/2 is also the limit as n gets large of the size of the rent [Pi][prime] that is available to the winner: [Pi][prime] = [Pi] - (n - 1)[x[prime].sub.i] = [n.sup.2][Pi]/(2[n.sup.2] - 2n + 1). As in rent seeking, with a large number of firms the combined expenditures dissipate nearly the entire rent available to the winner of a rent shrinking contest. These results are illustrated in Figure 1.

III. Collusion

Rent shrinking has interesting implications for the incentives for collusive behavior among the firms. In addition to the factors influencing collusion in a rent seeking framework, the size of the rent available to the winning coalition may increase as firms collude in a rent shrinking environment.

When k firms collude, they behave as a single firm. The number of effective competitors therefore falls to (n - k + 1) when a coalition of size k forms. The profitability of collusion (which may be negative) in the rent seeking game increases monotonically with coalition size k and decreases monotonically with the total number of firms n. In the standard rent seeking contest, profitable collusion by k firms is only possible for relatively comprehensive coalitions. Specifically, k-firm collusion is profitable in the rent seeking game iff n [less than or equal to] [k + [square root of k]], where [[center dot]] is the greatest integer function [1].(6) The gain to a colluding coalition arises from the coalition's ability to lower the aggregate rent seeking expenditures of its members. Firms that are not members of the k-firm coalition also gain from the collusion, therefore, as the aggregate expenditures of their competitors go down.

Collusion in a rent shrinking contest allows the colluding firms to reduce their collective expenditures, as the rent seeking game. But even if a colluding coalition does not change its total expenditures, the prize available to the coalition rises (holding the expenditures of non-colluding firms constant). There are fewer competitors, so the shrinking of the prize that results from the expenditures of competitors is attenuated.

In a rent shrinking game, k-firm collusion is profitable when the expected utility to a firm (the coalition) in a game with n - k + 1 competitors exceeds k times the expected utility to a single firm in a game with n competitors. The k-firm collusive profitability condition is therefore {[Pi]/(2[(n - k + 1).sup.2] - 2(n - k + 1) + 1)} - {k[Pi]/(2[n.sup.2] - 2n + 1)} [greater than or equal to] 0. The gains (or losses) from k-firm collusion, represented by the left hand side of the above inequality, are no longer monotonically increasing in k and monotonically decreasing in n.(7) Algebraic manipulation indicates, however, that k-firm collusion pays more often in the rent shrinking game than in the rent seeking game: k-firm rent shrinking collusion is profitable whenever n [less than or equal to] [k + [square root of (k - 1/4)] + 1/2].(8) For example, a 3-firm coalition can profitably form when n = 5 in the rent shrinking game, but is not profitable in the rent seeking game. Nevertheless, the coalition profitability condition is quite similar in rent shrinking and rent seeking contests. As the number of firms gets large, only coalitions that approach the size of the grand coalition find collusion profitable.

As with rent seeking, collusion in a rent shrinking environment benefits non-colluding firms, since the expenditures of their competitors decrease. Indeed, the external collusive gain to a non-colluding firm exceeds the collusive gain to any colluding firm, assuming that the colluding coalition splits equally its gain from collusion. Holding fixed their own expenditures, non-colluding firms gain with collusion both from a higher probability of winning and from the larger prize available to the winner.(9)

Collusion in a rent shrinking setting (as well as in the standard rent seeking framework) therefore displays some characteristics associated with public goods. As with other forms of public goods provision, firms may try to free ride in rent shrinking situations, by abstaining from collusion while encouraging others to collude. Incentives to free ride may be particularly difficult to overcome in rent seeking or rent shrinking contests, since nearly all firms must collude before there are positive gains to the collusive coalition. There is likely to be an "underinvestment" in collusion.

Table I presents some comparisons between the standard rent seeking game and the rent shrinking game, for the case when [Pi] = 1 and n = 3. Collusive gains to colluding firms are illustrated in Figure 2, and the collusive gains to non-colluding firms are depicted in Figure 3, for the case of n = 10 and with various sizes of the collusive coalition.

[TABULAR DATA FOR TABLE I OMITTED]

IV. A Generalization 

In some circumstances, the factors that reduce the rant going to a winning firm may simultaneously indicate that the expenditures of losing firms are not pure losses. It was argued in the Introduction, for example, that the winner of a patent race will get a prize that shrinks with its opponents' expenditures, since the opponents can innovate around the patent. These losing firms, therefore, are not out the entire amount of their expenditures, as they can still capture some rents.

A generalization of the rent seeking and rent shrinking games that captures the possibility of valuable expenditures by losing firms can be developed. Assume that a fraction of the expenditures of losing firms are recovered by the firms.(10) Let the fraction of expenditures recouped by a losing firm be denoted by (1 - t), with 0 [less than or equal to] t [less than or equal to] 1. In all other respects, the game is one of rent shrinking. Firm i then receives a payoff of ([Pi] - [summation of] [x.sub.j] where j[not equal to]i - [x.sub.i]) if it wins the rent shrinking contest, and receives a payoff of (-t[x.sub.i]) when it loses a contest. Firm i's problem is to choose expenditures [x.sub.i] to maximize [U.sub.i] = [p.sub.i] ([Pi] - [summation of] [x.sub.j] where j[not equal to]i - [x.sub.i]) - (1 - [p.sub.i])t[x.sub.i], where [p.sub.i] = [x.sub.i]/([summation of][x.sub.j] where j[not equal to]i + [x.sub.i]).

When t = 1, the losing firms get nothing for their expenditures, i.e., the game is a standard rent shrinking contest. As t decreases, a growing percentage of the losing firms' expenditures are returned. When t = 0, losing firms get all of their expenditures returned. Simultaneously, however, the rent available to the winning firm continues to shrink with losing firms' expenditures. Plugging t = 0 into the firm's objective function yields [U.sub.i] = [[x.sub.i]/([summation of][x.sub.j] where j[not equal to]i + [x.sub.i])][[Pi] - [summation of][x.sub.j] where j[not equal to]i - [x.sub.i]] = [p.sub.i][Pi] - [[x.sub.i]([summation of][x.sub.j] where j[not equal to]i + [x.sub.i])/([summation of][x.sub.j] where j[not equal to]i + [x.sub.i])] = [p.sub.i][Pi] - [x.sub.i]. So, when all expenditures are returned to losing firms, the rent shrinking game reduces to the standard rent seeking game. As t moves from 0 to 1, the contest moves from one of pure rent seeking to pure rent shrinking.

The symmetric Nash equilibrium in the generalized rent game is [x.sup.*] = (n - 1)[Pi]/[(1 + t)[n.sup.2] - 2tn + t]. The equilibrium expected utility for firm i is [Mathematical Expression Omitted]. As t increases, or as the contest approaches a pure rent shrinking game, equilibrium expenditures fall ([Delta][x.sup.*]/[Delta]t [less than] 0), and equilibrium expected payoffs also fall ([Delta][U.sub.i]([x.sup.*])/[Delta]t [less than] 0). The gains to a colluding coalition, however, are not monotonic in t. As t increases, there is more rent shrinkage, some of which can be saved by collusion. Simultaneously, however, the non-collusive expenditures are smaller, so there is less room to decrease expenditures via collusion.

The generalization of rent shrinking presented in this section may be of more than theoretical interest. Some real world situations appear to have elements of rent shrinking, as well as the refunding of expenditures. United States Department of Defense procurement of major weapons systems has some of the characteristics of a rent seeking game [8]. Furthermore, the research and development costs of the competing firms are at least partly reimbursed by the Department of Defense. The possibility of second sourcing indicates that the expenditures of losing firms may reduce the rent available to the firm that wins the initial production contract.

V. Conclusions

Rent shrinking occurs in many situations that previously have been modeled with a fixed prize. For example, in patent races or in contests for corporate control, it is likely that the expenditures of competitors reduce the value of the prize to the winner. This article has demonstrated that the phenomenon of rent shrinking lowers a firm's expenditures in seeking a prize and makes collusion profitable in more circumstances than in a standard rent seeking game. The rent seeking game and the rent shrinking game are both special cases of a game where the available rent shrinks with the expenditures of competitors, but losing firms are returned some fraction of their expenditures. If the fraction returned is 0, the game is pure rent shrinking; if all expenditures of losing firms are returned, the game is one of pure rent seeking. As the fraction of returned expenditures rises, a firm's expenditures and expected payoff also rise.

The idea to examine situations that involve shrinking prizes arose in a discussion at a seminar at Duke given by Barry O'Neill, particularly in the comments of Herve Moulin. Helpful comments were provided by Phil Cook, anonymous referees, and seminar participants at the Indiana University microeconomics workshop and the Southeastern Microeconomic Theory and International Trade Conference, Durham, NC, October 1993.

1. An example in the rent seeking literature is Rogerson [9]; in the patent race literature, Harris and Vickers [3]; in warfare, O'Neill [7]; and in campaign financing, Snyder [11].

2. Rent shrinking thus complements the analysis of [2], in which dead-weight losses tend to mute the political competition among interest groups for rents.

3. See Alexeev and Leitzel [1], Rogerson [9], and Tullock [13]. Other authors have examined situations where the value of the prize varies among contestants [4; 6]. Risk averse firms could behave substantially differently from risk neutral firms in rent seeking [10] or self-protection (negative "rents") [12] situations.

4. These results on the standard rent seeking model are contained in Rogerson [9]. It is assumed that the probability that any firm i wins equals 0 if all firms choose a zero level of expenditures.

5. Formulations of rent shrinking where the rent is reduced by alternative increasing functions of the aggregate expenditures of competitors produce similar qualitative results as the simple formulation used here. Similarly, typical variations on the standard rent seeking game produce no new insights. For instance, following Tullock [13], the probability that firm i wins could be given by [Mathematical Expression Omitted], where [Alpha] [greater than] 0. The optimal rent shrinking expenditures for firm i are then [Mathematical Expression Omitted]. (For [Alpha] [greater than] 1, there may be no solution, since depending on the exact value of [Alpha] and n, the second order conditions for a maximum may not be satisfied.) The "difference models" of Hirshleifer [5], alternatively, yield complex mixed strategy equilibria in symmetrical settings.

6. The formula applies to the case of "perfectly anticipated collusion," the only type of collusion considered here.

7. For example, when k = 2, the collusive gains are smaller (or rather, the collusive losses are greater) for n = 5 than for n = 6. Likewise, with n = 6, collusive losses increase as k goes from 2 to 3.

8. For every combination of n and k such that collusion by k firms is profitable in the rent seeking game (n [less than or equal to] [k + [square root of k]]), k-firm collusion is also profitable in a rent shrinking contest. Furthermore, collusion pays in some rent shrinking environments when it is unprofitable in the analogous rent seeking situation.

9. In optimally responding to the existence of collusion, the non-colluding finns will increase their rent shrinking expenditures.

10. The recoverability of bids is a more general issue in auction theory. In standard English auctions, there is "full recoverability," in the sense that losing bidders do not make any payment. Pay-all auctions such as the famous "dollar auction" are at the opposite end of the spectrum, with no recovery of bids [7]. The analysis in this section combines a rent shrinking setting with partial recovery of bids.

References

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