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Constitutional Political Economy, 12, 237 ± 253, 2001. 2001 Kluwer Academic Publishers. Printed in The Netherlands.

On Norms and Coordination Games: A Rent-Seeking Approach1 JOSEÂ CASAS-PARDO Departamento de EconomõÂa Aplicada, Universidad de Valencia, Av. Naranjos s/n, 4022 Valencia, Spain JUAN D. MONTORO-PONS Departamento de EconomõÂa Aplicada, Universidad de Valencia, Av. Naranjos s/n, 4022 Valencia, Spain

Abstract. The aim of this paper is to extend the rent-seeking literature to the equilibrium selection problem in competitive coordination games, i.e., games in which more than one equilibrium exists, and individuals' preferences are opposed. We analyze alternative correlated equilibria: contractual agreements and legally enforced equilibria. The latter are to be understood as the outcome of rents-seeking contests in which players invest resources in order to set a norm. The contest is analyzed in its basic two-person setting and later generalized to the two-populations case. There we show that the outcomes depend on the relative payoff structure of the game, the technological properties of the contest, and the population distribution. Finally, the efficiency analysis focuses not only on the extent of the rent dissipation, but also on the comparative analysis of the inefficiencies that arise in the market (not coordinated) equilibrium. JEL classification: C72, D72

1.

Introduction

Social systems are based, to some extent, on the coordination of individual decisions. There are circumstances when coordination arises spontaneously. Yet in other situations, coordination may not be reached because individuals fail to select a necessary action, usually due to the non-uniqueness of the equilibrium. In this case potential coordination failures may arise and institutional arrangements2 can be a way to prevent them. Any coordination situation implies an equilibrium selection problem which may be solved by various means, yet with one common feature: any attempt to explain the outcome of a coordination game must establish the mechanisms by which individual actions are focused on a strategy. In this paper, we submit that institutional arrangements are the mechanisms, formal or non-formal, that determine the outcome of social interaction. In some environments, one may expect individuals to adapt, through evolution and learning, to social constraints, thus minimizing the transaction costs generated in social exchange; simple rules of behavior or conventions arise as a response to the social setting.3 These solutions, fall into the category of the spontaneous order tradition,4 however, they do not exhaust all the non-formal possibilities. In fact, the equilibrium refinements literature5 has focused on how individuals can improve the outcomes of coordination interactions by using the structure of the underlying game in what has been called focal

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points. Finally, allowing the players to commit to a strategy by pre-play communication can be a contractarian way of eliminating the uncertainty that coordination situations imply. All these solutions share a common feature: they are non-formal in the sense that there is no explicit design of the institutional arrangement.6 Contrary to these settings, formal arrangements imply an explicit design with explicit aims: formal institutions have some defined ends and a set of means for their attainment. These prescribe the action to be taken by individuals every time they face the coordination situation, and there are sanctions for defectors. To some extent by setting legal norms, individuals modify the social constraints they face at the sub-constitutional level.7 This implies that the coordination framework is modified by means of the political decision making process. Formally the process will no longer be a coordination situation as individual strategies converge into collective decision making, which ultimately implies setting an equilibrium. Of course, we may think of the government as exogenous to this setting, with institutions created in a costless manner. However, it must be noted that if individuals are not indifferent on the action to be taken,8 they may engage in costly activities to affect the outcome of collective action. This paper develops a model of rent seeking within a coordination setup. Its main aim is to develop an alternative approach to a subset of rent-seeking contests by considering them as competitive coordination games, in which individual gains depend on the strategic interaction with the rest of players. Much of the literature on rent seeking focuses on regulations as the outcomes of rent-seeking games: a license to exploit a monopoly, a tariff to eliminate foreign competitors, etc. This paper addresses the regulation approach but within the context of coordination games. The paper is organized as follows. Section 2 analyzes two person competitive coordination games and alternative correlation devices to avoid coordination failures. Section 3 extends the rent-seeking approach to the correlated equilibrium literature by introducing political action as a costly correlating device, with rent seeking as a way of searching the solution of the game. The efficiency of the solutions is considered in section 4. Finally section 5 ends with some conclusions. 2.

Coordination Games and Correlation Devices

Let us first sketch the basic framework. Consider two individuals facing the decision making setup in Table 1.9 In this game, two players, row and column, have two strategies available, A and B. Both (A, A) and (B, B) are equilibria. Obviously, players benefit from choosing the same strategy, although individuals are not indifferent to the equilibrium. While column prefers B, row prefers A. We use the term competitive coordination games when referring to this subset of coordination games. Hence, players have a ranking of the equilibria but the competitive nature of the latter leads to an indeterminacy of the outcome, and the possibility of a coordination failure arising if players select opposed strategies. Coordination failures lead to efficiency losses, and are linked to individual beliefs about the game. In fact, these beliefs can be interpreted as a probability distribution over the set

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Table 1. A competitive coordination game. A A B

5 0

B 1 0

0 1

0 5

of pure strategies that in order to be consistent need be a randomization over the strategy set that is an equilibrium. Mixed strategy equilibria are consistent beliefs of the outcome of a game. In the game in Table 1 there is an equilibrium in mixed strategies in which players assign a positive probability to both strategies. In it, row plays {A, B} with probability 1={5/6, 1/6}, and column with probability 2={1/6, 5/6}. We can analyze inefficiencies in terms of the mixed strategy equilibrium payoffs. Individual mixed strategy payoffs are 5/6, which make collective payoffs 5/3. This is obviously less than the optimum amount corresponding to (A, A) or (B, B). Efficiency losses in this setting arise as coordination failures have a positive probability in the randomized strategy of both players. The literature on coordination games focuses on the means to improve the outcomes of these settings. Modifying the institutional setting can lead to efficiency gains by finding an arrangement by means of which both players agree on the equilibrium to be played. Departing from the concept of correlated equilibrium (see Aumann, 1974 and Aumann, 1987) individuals can implement a self-enforcing randomized strategy by using a correlation device which sends signals to both players to play a certain strategy. In this setting correlated strategies are to be understood as formal commitments on a given equilibrium or strategy subset. This includes the possibility of voluntary exchange between the players. Assume that individuals can agree to play both A or B with probability 1/2. In this case, the extended game includes an agreement option shown in Table 2, that yields collective Pareto optimum outcomes. If both agree, then a correlation device, which can be understood as a mediator, signals to either play A or B. If only one player agrees, she follows her most preferred strategy. The agreement option is to be understood as a contract, and it is open to the players either to sign it, and follow its prescriptions, or not to sign it, and be subject to the indeterminacy of the setup.10 In this example contracts are an efficient and (probably) costless way of coordinating individuals and avoiding welfare losses. Of course Table 2 shows one of the infinite possible contractual agreements as any correlation device that secures any player payoffs over her security level could be agreed. Thus, we may find an indeterminacy in the game in the case of a contractarian solution, as, without any initial constraints, any of the infinite solutions could become an outcome. Additionally, limits to the voluntary exchange arise from the nature of the setting, and more specifically from the transaction costs that agents face, which may, when

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Table 2. A competitive coordination game with a contract option (C). A A B C

5 0 5

B 1 0 1

0 1 0

C 0 5 0

0 1 3

0 5 3

positive, limit the extent of voluntary exchange. We may think of a correlation device which emits costly signals to the players. Individuals may be willing to pay for them as long as the costs do not exceed the gains above the reservation payoffs of each player. In this setting, contracts are exogenous means to attain coordination. Alternatively, we may think of an endogenous correlation device. In this case, players investing resources can determine the outcome of the game. As we noted above, a mediator which points to a strategy with probability 1/2 is only one of infinite possible options. Any rational agent could devote resources to affect the mediator so as to make it point toward her preferred option with a probability higher than 1/2 as long as the amount invested in the process does not exceed the expected gains. In fact, introducing a norm in the game that leaves just one legal strategy (be it A or B), is like introducing a correlation device that sends signals for playing a strategy with probability 1. In this case, the institutional arrangement emerges as an outcome of the political process. Norms and contracts have that, once in existence, individuals are forced to follow them. However, they also differ in many aspects. Contracts, as correlation devices, are self-enforcing as no player is interested in deviating from it. By contrast, the coercive nature of norms is enforced by the monopoly of the state through the legal system and punishment schedules for those breaking them. While individuals may voluntarily sign a contract, norms are the outcome of collective decision making, and thus not subject to voluntary exchange processes. Many real world institutions are the outcome of norms which define them as the way a game is played when multiple equilibria are available.11 In the example of game 1, a norm to play A would yield (A, A) as the only equilibrium solution of our game. In this case, and as we already pointed out, there would be a redistribution of welfare among the players: in our simple setup, after introducing the norm, there would be a redistribution of welfare toward row. Next we analyze this situation. 3.

Rent-Seeking Solutions to Competitive Coordination Games

Devoting resources to set a correlation device may be needed in case of positive transaction costs (costly contracts) or in a collective action setting. This paper addresses

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Table 3. A generic competitive coordination game: , , ,  > 0,  > ,  > . s2

s1 s1 s2

 0

0

0

0 

the latter. Rent-seeking contests are characterized by the investment of resources in order to affect the outcomes of political processes, and get a benefit or rent. This introduces the possibility of the players modifying the setting endogenously. In the next subsection we analyze the two-player case. 3.1.

Analysis of the Two Player Contests

Let us start by analyzing the rent-seeking solution departing from the setting in which both agents face a symmetric payoff structure. Consider the game in Table 3 where we allow the emergence of collective action. In this case players can engage in rent-seeking activities in order to alter the outcomes. Let x1 and x2 be the amount of resources that row and column allocate in rent-seeking activities, to be understood as the individual efforts in the competition. Let Pi, agent's i probability of winning the rent or obtaining the norm, be a function of these resources. In its logit form, with symmetric and risk-neutral players, the original formulation found in Tullock, 1980, is Pi ˆ

xri : xr1 ‡ xr2

…1†

The efficient level of rent seeking arises as the solution of maximizing the expected value of the game, which is given by V e ˆ Pi Ri xi ; with Ri being player i's rent, which is the payoff of the preferred equilibrium. As will be shown, expected gains from setting an equilibrium will play a key role in explaining the outcomes of the game. Assuming a symmetric game the aggregate amount of resources invested depends exclusively on the parameter r of expression (1), being the solution of the problem given by xi ˆ

rR : 4

…2†

As xi is the optimum amount of rent seeking of agent i, in the two-player setting, the total amount of rent dissipation will be given by the summation over i (that is 1,2) of expression

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(2). As noted above, the overall amount of rent dissipation will depend on r, and we can see that it is less than the rent for r < 2. With r > 2 over-dissipation holds. In the case of asymmetric payoffs, with R1 6ˆ R2, the individually optimum amount of rent-seeking activities for player i, is given by xi ˆ r

r Rr‡1 i Rj

…Rri ‡ Rrj †2

;

…3†

with i 6ˆ j. Now the investment in rent-seeking activities depends on the technological characteristics of the competition, the parameter r, and on the payoff structure of the coordination game. In order to simplify the analysis we express one rent in terms of the other. Suppose that R1  R2. Then12 R2 ˆ aR1 with a 2 …0; 1†;

…4†

and expression (3) could be written as x1 ˆ rar

R2r‡1 1

…Rr1 …1 ‡ ar ††2

:

…5†

And the rent dissipation ratio13 is Dˆ

ar …1 ‡ a†r …1 ‡ ar †2

:

…6†

From expression (6) we derive the following result: Proposition 1 Let the competitive coordination game in Table 3 define the payoff structure of a two person rent-seeking contest. Then, for any given r, the rent dissipation increases with a, and reaches its maximum for a = 1. Proposition 1 yields a misleading conclusion with respect to the resources that will be invested in the game. Now both players face different potential gains, and the parameter a depicts this fact. As the rent dissipation increases with a, this leads to the unequivocal conclusion that the symmetric game is a benchmark for the rent dissipation: for a given r, the rent dissipation X is increasing with a, with its maximum in the symmetric case. The sharpness of this result contrasts with the predictions with respect to r: equation (5) shows how the returns of the competition affect the extent of the rent-seeking activities. However, the sign of the variation is ambiguous and depends on the parameter a. Only in the symmetric case, i.e., a = 1, the rent dissipation is increasing in the parameter r; in the general setup, it varies negatively with r for highly asymmetric games, and positively for the rest of values of a. In order to get the intuition of this solution, we resort to its graphical representation. Fig. 1 shows the relation between the rent dissipation ratio and a, for some r-values

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Figure 1. Rent dissipation in an asymmetric rent-seeking competitive coordination game (constant and increasing returns).

(constant and increasing returns). In it rent dissipation, D, is increasing with a in all cases. The graphic shows the varying relationship between the parameter r and D, proving that in the non-symmetric game, for a given a < 1, the rent dissipation is maximum for a finite value of r. Similar qualitative results are obtained for decreasing returns. Summarizing, we may state that for any two r-values r0 and r00, with r0 < r00, there is a value a < 1, such that for a < a then Dr0 > Dr00, and for a > a then Dr0 < Dr00. We can state that symmetries play an important role in determining the amount of rent dissipation: for highly asymmetric games rent dissipation keeps at low levels, even for high, and perhaps unlikely, r values. Conversely, for high a values, there is an almost direct relationship between the rent dissipation ratio and the returns of the rentseeking game. The higher the returns level, the more sensitive becomes the rent dissipation with respect to small variations in the relative payoffs structure of the game. Hence, slight modifications of a imply significant increases in the extent of the total outlays. We can summarize the previous analysis as follows: considering the symmetric case as a limit, for a wide range of returns and rent structures, specially for those related to a parameter values ranging from low to moderately high, under-dissipation will be the main result. The meaning of ``moderately'' should be taken cautiously. Yet, one should consider that the higher the inequality of the game, the larger the range of returns that can be considered to fall within this result. Moreover, for a large range of game payoff structures the returns that yield over-dissipation are highly unlikely.

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3.2. Collective Rent Seeking Although the interaction between two players may be of interest for analysing the basic setting, collective action implies a group of n > 2 individuals. In this case the extension of the problem to a binary population of n individuals is straightforward, considering Table 3 as the outcomes of the interaction of two opposed preferences class of individuals or social groups. Note that we are analyzing two populations, so the framework we use is that of a binary choice (in our example either A or B) and not of n different alternatives. We will refer to population or group A, as the one composed of row individuals in the basic coordination game. In a similar way group B will be that of column players. Here, in addition to the coordination problem, we should consider those problems related to group formation and individual contributions toward the attainment of the objectives of the social group. These are not independent of the chosen modeling framework.14 Let us consider the general setting with n = n1 + n2 players, n1 being the number of individuals in group A, and n2 the number of individuals in group B. Let q be the proportion of type A players in the population. In this case the probability of interacting with a type B player will be 1 q. If we consider n individual games, then the payoff for the i-th player of group A will be V1ie ˆ P1 …1 q†R1 x1i ;

…7†

with P1 ˆ

X X X1r ; being X1 ˆ x1i ; and X2 ˆ x2i : r r X1 ‡ X2 i i

Similarly, we may define the payoffs of an individual of group B as V2ie ˆ P2 qR2 x2i :

…8†

Solving the game for the n players, by using symmetry, we get the following solutions x1i ˆ rR1 x2i ˆ rR1

ar …1 q†r‡1 qr n1 …pr ‡ ar qr †2

;

ar‡1 …1 q†r qr‡1 n2 …pr ‡ ar qr †2

…9†

:

…10†

From (9) and (10) we get the following results: Proposition 2 Consider the collective rent-seeking game with n players facing a binary election. Then, (i) Individual effort levels are decreasing with the size of the group (and of the population).

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(ii) Let # = n2/n1; then the ratio of individually optimal effort levels, i.e., x1i /x2i , is #2/a. (iii) The ratio of collective effort levels, i.e., ix1i / i x2i , is #/a. In this setting, as the probability of getting the prize increases with the aggregate efforts, groups face a social dilemma. In fact, result 2-(i) shows how individual rent-seeking expenditures fall with rising group size,15 which shows that, other things equal, the group size affects individual efforts. This is due to the positive externalities that the expenditure of one member of the group has on the rest of the members. Since any individual effort increases the winning probability for the whole group, individuals may free-ride on contributions by others. Result 2-(ii) compares efforts made by individuals of each group, which depend on the relative size of the groups and of the rents. The greater the relative size of a group, the lower its individually optimal contribution; the higher the asymmetry of the rent (the closer a to zero), the higher the efforts of an individual of group A compared to an individual of group B. Depending on the relative value of a (or #) we may state when the individually optimal effort of one group will be larger (smaller) than the other. In fact x1i > ( < )x2i , if a < (>)#2. Finally result 2-(iii) shows the aggregate efforts ratio. Given that the winning probability is determined by the relative efforts of each group, it is interesting to analyze it. As with individual efforts, collective rent seeking depends on the relative size of the groups and the rent. In the two players individual game, the probability of getting the rent was entirely defined by the size of the rent, hence the rent was allocated (with higher probability) to the player that values it most: row gets the rent with higher probability given that a 2 [0, 1]. By contrast, group rent seeking allows for alternative solutions. Now the probability of winning the rent also depends on the relative size of the group. It could be the case, that group B wins the rent if a > #. Combining both parameters (a and #) two alternative situations emerge. (i) The larger group has a larger valuation of the rent, # < 1. In this case there are three possible qualitative equilibria. One in which the larger group gets the rent (# > a and X1 > X2), one in which both have equal probability (# = a with X1 = X2), and one in which the smaller group gets the rent (# < a with X1 < X2). (ii) The smaller group has a larger valuation of the rent, # > 1. In this case the only solution is the small group getting the rent (# > a and X1 > X2) with a higher probability. These conclusions show how there is no a priori expected outcome on the basis of size or expected rent of a social group. In fact, a minority which values the rent less may end up enforcing an equilibrium, depending on the structural parameters of the interaction.

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Rent Dissipation in the Collective Contest

Another interesting aspect is that of the aggregate amount of rent dissipation. With collective rent seeking, the rent dissipation ratio is given by Dˆr

ar …1 q†r qr 1 ……1 q† ‡ aq† n……1 q†r ‡ ar qr †2

:

…11†

D may be less than or greater than one. Nevertheless, we note some facts with respect to D. First, D is decreasing in n, thus rent-dissipation will tend to fall, as per proposition 2, with the population size. Second, for decreasing and constant returns, i.e., r  1, @D/@a is positive, while @D/@q is negative. The first result agrees with our previous analysis. The second shall be further explained. Parameter q is to be understood as the relative size of the population with the higher valuation of the rent (population A). As it has been shown, outlays will be minimum for this being the largest possible group, in case of q = 1, and will increase as the relative size of this group diminishes. For increasing returns, r > 1, both derivatives have an ambiguous sign. It is interesting to note that the sign of @D/@q changes for a given value q . It is also for this value that D is maximum. The value q varies with the parametric set, and specifically with a. In fact q decreases for increasing a values, which leads to the following conclusions: Proposition 3 For all r > 1, there is a value q such that (i) for every q < q rent-dissipation is increasing in q, and for every q > q rent-dissipation is decreasing in q; hence, rent dissipation is maximum at q . (ii) The boundary value q is decreasing in a and increasing in r. Rent-dissipation is now determined by the parametric set. From result 3 we note first that q points to a population structure which maximizes the rent dissipation. This maximum rent dissipation will be unambiguously increasing in a only if r = 1 and q ! 0. For other parametric structures, according to our previous results, it will only be increasing if q < q . More symmetric games imply more rent-dissipation. In general, there is a combination of rent valuation and population structure that yields the highest rent-seeking outlays, for a given rent-seeking technology. Second, quantitative modifications of this, i.e., modifying the size of the increasing returns, play a key role in determining the efficient allocation of rent-seeking expenditures. In fact these are increasing with the returns. It should be noted that, given the dependency of the ratio on the parametric set, under-dissipation is the most likely outcome. This is strongly supported for increasing population sizes. 3.2.2.

Pareto Optimum Group Rent Seeking

Finally we conclude this section analyzing the Pareto optimal allocation of group rentseeking expenditures. It occurs for the collective maximization problem

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V1e ˆ P1 …1 q†n1 R1 X1 ;

…12†

V1e ˆ P2 qn2 R2 X2 :

…13†

The individual (Pareto optimum) solutions that arise from this problem are xp1i ˆ rR1 xp2i ˆ rR1

ar‡1 …1 q† …1 ‡ ar †2 ar q …1 ‡ ar †2

;

…14†

:

…15†

In this case outlays are, as expected, larger than those corresponding to the Nash equilibrium of the game. In fact the quotient of Nash to Pareto equilibrium values are x1i =xp1i ˆ x2i =xp2i ˆ

…1 ‡ ar †2 …1 q†r qr n1 ……1 q†r ‡ ar qr †2 …1 ‡ ar †2 …1 q†r qr n2 ……1 q†r ‡ ar qr †2

;

;

which are less than unity under mild restrictions on the parametric set.16 Note that in case of group-optimal outlays it is the group with a higher valuation of the rent that gets the prize. In fact the quotient of collective optimum outlays between groups is given by X1p 1 ˆ ; X2p a

…16†

which obviously is greater than unity if 0 < a < 1. Thus, in this case the probability of getting the rent is related to its valuation, and is independent of the group size. As a consequence, rent dissipation is larger than in the Nash case. Note that would it be possible to induce individuals to show optimal behavior, then the problem would be again that of the two-player setting, in which the one who puts a higher valuation on the prize, obtains the rent.

4.

Efficiency of the Rent-Seeking Correlated Equilibrium

In the previous analysis we have addressed the question of the efficiency of the rentseeking contests by means of the dissipation ratio. From this perspective we measure the allocative efficiency of a non-market process by the resources that participants invest to set a norm in terms of its valuation. While this may be the right approach in most collective contests, it ignores some important aspects of the framework we are developing. In fact, as

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the market allocation may fail to reach an optimum in case of a coordination setup, agents may be facing costs, as in the mixed strategy equilibrium coordination failures are given a positive probability17. Due to this fact, the efficiency of non-market allocation should be compared to the efficiency of the costly correlated equilibrium solution when no device is imposed and individuals may be subject to coordination failures. Based on this, next we propose an alternative approach to measure the efficiency of a rent-seeking correlation device. First we will analyze the two person game. Then we will consider collective rent seeking. 4.1. The Two-Person Case Coordination failures arise as agents place positive probabilities on both equilibria. In fact, given the game in Table 3, the mixed strategy equilibrium for row and column is {/ ( + ), /( + )} and { /( + ), /( + )} respectively. Then the expected value of the game is given by U1e ˆ

1 R1 1 ‡ 1

…17†

U2e ˆ

2 R2 ; 1 ‡ 2

…18†

where i is the ratio of equilibrium payoffs, i.e., it is the result of dividing the preferred equilibrium payoffs by the alternative equilibrium payoffs, / and / respectively. Obviously, i < 1. We also know the overall individual rent-seeking outlays xi , given by expression (3). In this case we can observe that !i = xi /Uie, is a measure of the individual rent dissipation in terms of the alternative market solution. To put it differently: let i /(1 + i), that is Uie/Ri, be a measure of the relative inefficiency of the market solution, and let Di = xi /Ri, the individual rent dissipation, be the relative inefficiency of the rent-seeking solution. Then (xi /Ri)/(Uie/Ri) = xi /U ie will give a measure of the efficiency (inefficiency) of the collective action solution with respect to the market solutions. In that case, 

if !i > 1, the rent-seeking solution is, for individual i, more inefficient than the market outcome;



if !i < 1, the rent-seeking solution is, for individual i, more efficient than the market outcome. In this setting the individual efficiency ratios are given by !1 ˆ r

ar …1 ‡ 1 † …1 ‡ ar †1

…19†

!2 ˆ r

a…1 ‡ 2 † : …1 ‡ ar †2

…20†

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Expressions (19) and (20) show that the relative individual efficiency of the rent seeking solution depends on the parametric set of the game (a, i), and the technological conditions of the contest (r). The technological nature of the rent-seeking competition affects the efficiency outcomes by modifying the parametric set that leads to efficient (inefficient) solutions for both players. Define the set of efficient parameter combinations (a  i) as the one that leads to an efficient outcome by a rent-seeking correlation device, i.e., !i < 1, then the dimension of this set is decreasing in r. Fig. 2 shows that a costly correlation device is more likely to be efficient the lower the returns of the rent-seeking activities. Note that there is an asymmetry in the outcomes for both players: The solid line sets the limit of the efficient solutions for player 1, while the dashed line does the same for player 2. Over this line, !i < 1. In case of increasing returns the efficient set is larger for player 1 (areas marked as I and II) than for player 2 (area marked as I). In case of constant returns, both players face the same sets. Finally, for decreasing returns player 2 faces a larger efficient set (I and II) than player 1 (I). 4.2.

Efficiency of the Solutions in the n Person Setting

Now we consider the n individuals binary choice game. In this case we have !1 ˆ r !2 ˆ r

ar …1 q†r‡1 qr 1 …1 ‡ 1 † n……1 q†r ‡ ar qr †2 1

;

ar 1 …1 q†r 1 qr‡1 …1 ‡ 2 † n……1 q†r ‡ ar qr †2 2

…21†

:

…22†

Obviously, as per the conclusions in the previous section, the population size n affects (decreases) the cost of rent seeking, thus making this a more efficient solution as group size grows. In addition to this, there are several other elements to be taken into account. First, the distribution of the population benefits the larger social group, giving it an efficiency advantage. The more unequal the distribution of the population, i.e., either q ! 0 or q ! 1, the more efficient the larger group is in rent seeking. Second, the subjective valuation of the rent also plays a key role, reinforcing (or counterbalancing) the benefits from the population distribution, as is expected. Hence, other things being equal, an increase in the subjective valuation of the rent increases the efficiency of a group. However, the population distribution dominates this fact. Hence we find that 1. for q > 0.5 (and by assumption a 2 [0, 1]), the large group has an efficiency advantage, i.e., !1 < !2; 2. for q < 0 there is a payoff structure a 2 [0, 1] such that for any a < a the smaller group is more efficient for a wider range of parametric values. Conversely for a > a , the situation is reversed. Finally the technological properties of the contest affect slightly the previous qualitative conclusions. As r tends to 0, a ! 1, which means that in the limit, under decreasing

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Figure 2. Parametric combinations and efficiency of rent-seeking solutions: top r = 3, bottom r = 3/4.

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returns, the smaller group has an efficiency advantage with independence of its rent valuation. On the other hand, as r increases, a ! 0, which implies that, under increasing returns, it is the larger group which has the efficiency advantage. 5.

Discussion and Conclusions

Many economic and social interactions can be analyzed within the framework of coordination games. Alternative technological standards or the choice of the dominant language in bilingual communities are two examples in which coordination issues may arise in a social setting. Possible solutions may be analyzed from different standpoints. The literature on coordination failures has been fruitful in examining alternative means for avoiding failures. The present paper has explored the possibility of rent seeking as a costly correlation device. It also offers a comparative analysis with alternative private solutions: market and contractual agreements. While market solutions imply coordination failures as there is no means by which individuals can agree on the solution, contractual and rentseeking solutions avoid this problem by means of correlated strategies that coordinate individual behavior. The main objection to a contractual solution in this setting is that it leaves the outcomes undefined, as there are many possible outcomes (theoretically infinite) of the negotiations. By contrast, collective action by means of rent seeking leads to an endogenous (probabilistic) outcome. Collective solutions, characterized as the outcomes of rent-seeking games, imply the redistribution of resources in a social system. Departing from the two-player setting, we extended our analysis to a collective rent-seeking scenario in which externalities, the valuation of the rent, and the technological conditions of the contest drive the outlays. A direct consequence is that now the rent may be allocated to any group in contrast with the sharp two-player game results. It has been shown under which circumstances minorities or majorities set the political outcome. Finally note that the competition that arises is but an alternative means for allocating resources. In fact, the waste of resources implied in the rent-seeking contest can be compared to the inefficiencies of market solutions. From our analysis we were able to conclude that rent-seeking solutions are (in general) individually efficient for a wide range of parametric sets and technological conditions. This implies that individuals by opting for a costly correlation device avoid further resource waste. Notes 1. The authors wish to thank two anonymous referees for their helpful comments and valuable suggestions on an earlier version of this paper. The usual disclaimers apply. 2. Thinking in game-theoretical terms, institutions are the social outcome of coordination games. From this point of view institutions, in a broad sense, are to be understood as the means for selecting one of the possible equilibria. 3. There is a large literature on conventions and their economic implications; some references are (Goyal and Janssen, 1997), (Sudgen, 1989), (WaÈrneryd, 1990), or (Young, 1996). However, works on conventions are

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closely related to the literature on evolution and learning, and unbounded rationality approaches to games; here (Binmore, 1990), (Conlisk, 1996), (Crawford and Haller, 1990), (Mailath, 1998), (Nelson, 1995) or (Vanberg, 1986) are relevant. 4. Its roots being the Austrian tradition and especially the work of Hayek (see for example Hayek, 1985). 5. See (Harsanyi and Selten, 1988). 6. One may doubt whether a contractual agreement to follow a strategy is a non-formal institution. To some extent there is a design by the various players to commit to one strategy. In this sense it may be considered a formal arrangement. However, in most of the game theory literature pre-play communication is a verbal type of commitment that shares some characteristics with non-formal arrangements, and hence it should be considered one. 7. Of course this will change the setting from a coordination setup to a different setup. However we are not concerned with the output but with the redistributive effects of this mechanism, which can be analyzed in a similar way than welfare losses in a rent-seeking process. 8. And this is the case that we will be discussed later, in this paper. 9. To simplify the notation we will consider that off-diagonal payoffs are zero. Including non-zero off-diagonal payoffs will not modify in an essential way the rest of the argument, as long as (A, A) and (B, B) remain as the only equilibria of the game. 10. It should be noted that the randomized mediator is but a theoretical trick to make an equitative redistribution of the payoffs between the players as the expected utility of following it is 3 for both players. 11. To some extent these are norms-enforced institutions in contrast to non-formal institutions like conventions, which emerge in an evolutionary fashion. 12. Obviously if this was not the case we need only re-arrange the matrix in Table 3 to do so. 13. The rent dissipation ratio is the overall rent dissipation x1* + x2* divided by the rent. As we may have two alternative measures of the rent, R1 and R2, in this paper we consider the rent dissipation ratio in terms of the largest R1. 14. In fact the problem can be tackled in different ways; for simplicity, and to maintain the consistency with the basic two-player problem, we have considered two alternatives. The first is to think in terms of n individual coordination games. In this setting players of one type face a coordination game with some probability with a player of the opposite type. In this case an individual gets the rent if the correlation device favors her group. Individual efforts in the rent -seeking competition affect the probability of the group to get the rent. The second alternative is to think in terms of a collective contest of n players in which the rent is to be divided among its participants. This implies a rule for distributing the rent among the participants of the winning group. Given the description of the setting we consider it should be a proportional rule, which endows each player in the group with a 1/n share of the rent. Both environments yield similar qualitative conclusions. This is why we only consider the first alternative. 15. Obviously, they are also decreasing in the total population size n; we just have to substitute n1 = qn in expression (9) and n2 = (1 q) n in (10). 16. Note that as group size increases it enlarges the denominator, thus diminishing the quotient. 17. We thank an anonymous referee who suggested the development of this idea.

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