Noise in rent-seeking contests

Noise in rent-seeking contests

The analysis of a new contest success function

‘Doctoraal’ Thesis Econometrics Faculty of Economics

University of Groningen

Abstract

Contents

1. Introduction 1

2. Contests and their success functions 3

3. Stackelberg rent seeking with the new contest success function 23

4. Collective rent-seeking contests 48

5. Conclusion 60

1. Introduction

Contests are a very common phenomenon in our every day lives. Whether it is watching a sports game on television, voting for a new government, reading an advertisement in the paper, getting promoted to a better job, companies competing for a natural monopoly,

litigation, or even countries waging war, competitions are everywhere. In all competitions, the contestants expend effort to increase their chance of winning. This effort could be time spend on training in order to become better at sports, or money spend on lobbying, or increasing the defence budget.

Since contests are so various, and the efforts spend in competitions so large, it is rather surprising that economist only started studying it a few decades ago. The contest theory started with Tullock questioning what the real social cost of monopolistic markets were. He argued that the lobbying expenditures made by potential monopolist in order to obtain the privileged position in the industry should be seen as social cost as well. After Tullock’s pioneering work, the field of contest theory started expanding rapidly in the 1980s, as it became clear the theory was far more reaching than the original natural monopoly setting. The field of economics that studies contests has since then become known as rent seeking, and it has subsequently become an important part of modern game theory.

Not only did the rent-seeking literature focus on the numerous applications both within and outside economic theory, but also the design of the contests itself has become a topic of great interest. Nitzan (1994) surveyed the alternative ways of modelling rent-seeking contests. He focussed on the relationship between social waste associated with the rent-seeking process and the underlying characteristics of the game, e.g. the number of players, their attitudes towards risk, the asymmetry among the players, the source of the rent, the nature of the rent or the rent setter.

Other surveys of the rent-seeking literature, for instance Tullock and Lockhard (2001) or Konrad (2007), show that the debate about the strategic aspects of rent-seeking contests is still very much alive and still rapidly expanding.

One of the underlying characteristics of the contests still being studied, is the relationship between the efforts put forward by the contestants, and the chance that they will win the competition. The mathematical representation of this relationship is known in the rent-seeking literature as the contest success function. Starting with Hirshleifer (1989) and Skaperdas (1996), papers have been written questioning whether Tullock’s original contest success function is the most suitable one. Since this function assigns the winning probabilities to the different players, depending on the efforts they put forward, it is one of the most essential parts of every contest.

My thesis is organised as follows: chapter 2 sets out a brief overview of the rent-seeking literature, explains the basic model of a contest, and introduces the new contest success function. In chapter 3 as well as in chapter 4, I will discuss a well-known model from the rent-seeking literature, and subsequently introduce the new contest success function to these models. That is, I will examine how luck as a factor in the players’ decisions will change the outcomes of two well-know models in the rent-seeking literature. In addition, I will

2. Contests and their success functions

Modern welfare economics has paid considerable attention to the understanding of the losses associated with market imperfections. The earlier analyses of the deadweight losses from monopoly, taxes, tariffs, trade barriers, etc. were of a static nature. They described and measured the welfare losses of imperfect markets without questioning how particular imperfections had arisen. In the 1960s, Gordon Tullock argued that the resulting welfare losses are considerable larger than the standard deadweight losses in the literature on welfare economics. According to Tullock the market imperfections should not be seen as static, but rather as the result from systematic efforts by the groups affected by those markets. Therefore the efforts put forward by the groups benefiting from for example a monopoly or trade barrier should also be considered as a part of the welfare cost.

Tullock (1998) himself describes that a conflict between his instinct and his observation triggered him in writing his paper on welfare losses in the 1960s. According to the economic literature at that time the social cost of a monopoly was the deadweight loss triangle which gives the loss of consumer surplus resulting from the higher price a monopolist charges (compared to the price in a market with perfect competition). The profits of the monopolist were not considered as a welfare loss, but rather as a transfer of welfare from consumers to the monopolist. Tullock also observed that the economic theory at that time assumed that a monopoly could be obtained without any cost at all.

His instinct however suggested that he should shift his attention from the welfare triangle to the profit rectangle (Tullock, 1998, p. 1). Surely, it should be expected that potential

monopolists would go through considerable effort to obtain these monopoly profits. Rational entrepreneurs should be willing to invest large sums in order to obtain a monopoly position and its accompanying profits. In a perfect situation, they would be willing to invest the same amount as they would expect to gain from the monopoly. Therefore, the efforts entrepreneurs make could be as high as the rectangle showing the monopolist’s profit. The welfare losses should not only consist of the deadweight triangle (as in the standard literature), but also the profit rectangle representing the monopoly rents. Tullock shifted his attention to the size of the investments potential entrepreneurs would make in order to gain the monopoly rents. That is, he started to look at the rent-seeking expenditures made by potential monopolist and the accompanying social losses, rather than focussing solely on the deadweight losses that arise after the monopoly has been established.

Although the early rent-seeking literature mainly focussed on measuring losses associated with interest groups trying to acquire profits from monopolies, taxes, tariffs and trade barriers, the idea that resources are consumed in competitive contests is, however far broader than the pioneering papers suggest. The rent-seeking logic has been applied to general problems in political science, history, sociology, anthropology, biology, and philosophy, and its core models have been generalised to form an important strand of modern game theory (Congleton, 2006, p. 1)

2.1. The literature on rent seeking

rent-seeking contests, and the research on various applications and case studies which use the rent-seeking theory.

The mathematical foundations of the rent-seeking contest and the different possible contest structures have received considerable attention. The early research mainly focused on the question whether the aggregate investments made by all players in the contest would equal the size of the rent. Tullock originally thought that the aggregate efforts would be almost equal the contestable rent. Later he discovered that the proportion of the rent dissipated depended on the number of players participating in the game. Only when the number of players in the contest is sufficiently large, the prize will be completely dissipated.

Nitzan (1994) continued the research on the relationship between the characteristics of the contest and the extent of rent dissipation. In his survey, he shows that due to risk aversion of the rent seekers, heterogeneous rent-seekers, uncertainty about the prize, complete rent

dissipation is not to be expected. Nitzan also did further research on how rent-seeking contests are affected by the interaction among the rent-seekers. For example he investigated the nature of the competitors, and questioned what the results would be if the rent-seekers were groups of individuals. He showed that modelling group rent seeking reduced the dissipation rate as well.

The type and number of prizes won in rent-seeking contest similarly affects the extent to which recourses are invested in those games. In much of the contest literature, the prize is assumed a newly created private good. However, the prize may also be a private good transferred form one group to another (which may lead to the losing group organising

themselves and resisting the transfer). In addition, the prize could also be a public good (e.g. a publicly funded park, or pollution removal by the local authority), which would affect the efforts put forward differently yet again. A public good may lead to free-riding behaviour of the rent-seekers, and thus lower the total recourses invested.

Another example of how the analytical research on the specifications of the model has extended the Tullock model in many different directions, is the exploration of dynamic rent-seeking models. Instead of using static games, which are solved by using the Nash

equilibrium solution concept, models in which each player can react to the efforts put forward by other players. Leininger (1993) studied a two-player, two-stage contest in which a player’s strategy consists of two components: the timing of his move, and his rent-seeking effort. He shows that the players will always agree on the order in which they will move: the weaker player will always move first.

Similar to the literature focussing on the analytical research, the rent-seeking literature focussing on empirical case studies and applications of the rent-seeking theory has also expanded rapidly in the last decades (Congleton, 2006, p.6).

Part of the literature focuses on estimating empirically how large the welfare losses associated with trade-barriers and monopolisation are, and shows that the losses are considerably larger than in studies which do not take rent-seeking expenditures into account. Another part of the literature focuses more on the understanding and on the motivation of investments made by politically active interest groups. It shows how the contests in lobbying for rents are

organised, or even suggests how they could be organised better.

conflict over the allocation of resources and the associated rents resulting from different players trying to get control over these resources, have reduced the speed and efficiency of the transition (Congleton, 2006, p. 14).

Another example is promotional competition in which firms try to increase their market shares by advertising and other marketing activities (Konrad, 2007, p. 6). The investments made by a firm to persuade customers to buy a particular product, like paying for

advertisements in newspapers, making TV spots or hiring sales agents are mainly made up-front. Since other firms selling that same product will do the same, these investments in marketing activities could be seen as efforts in a contest in which the effort choices determine the market shares of all firms. Konrad (2007, pp. 7-8) points out that these efforts are

substantial: marketing expenditure may be as high as 30 percent of total sales revenues in certain sectors.

Beauty contests are also an example that Konrad uses to show how much money is involved in contests. The choice of locations for the Olympic Games, for instance, is a contest in which the players will expend considerable effort in order to win the prize (being allowed to host the Games). Konrad (2007, p.11) mentions that Sidney spend 25.2 million Australian dollars to convince the International Olympic Committee that it was the best place for the Olympic Games of 2000.

Even non-economic phenomena can be studied with the rent-seeking model designed by Tullock. The way in which the court system is organised has remarkable similarities with the Tullock-contest. Both parties in a lawsuit can increase their likelihood of winning by

expending more effort, by paying more money to hire more and better lawyers. However, when the other party increases his efforts as well, the chances of winning the law suit (the chances of winning the prize) might not increase by much. The only consequence is that as well the plaintiffs as the defendants are wasting their money on legal expenses, because the final result, the chances of winning the lawsuit does not change.

In addition to these examples, the rent-seeking theory has also been used in labour-conflicts, election campaigns, industrial disputes, sports, education and even war. Since the amount of economic and social settings in which investments by rivals affect the success of their individual enterprises, and in which depending on the particular settings of the contest, considerable resources can be wasted, are so numerable, it is important that the rent-seeking model is based on firm analytical ground. As will be seen in the remainder of this chapter the way in which a rent-seeking contest is modelled is not unambiguous.

2.2. The basic Tullock model

Therefore Tullock proposed a model, in which n rent seekers compete for a prize (rent) of size (value) V. If a player i makes an expenditure of xi in order to capture the rent, his probability of success is assumed to be given by

≠ + = i j r j r i r i i _{x x} x p (2.1)

where xj denotes the simultaneous expenditure by his rival j. If all the players decide not to invest (i.e. if xi = 0 for all i), Tullock defined the winning probabilities of the players to be equal (i.e. pi = 1/n for all i).

The parameter r in the expression above reflects the marginal return to the lobbying

expenditures. That is, the higher is r, the more sensitive is the probability of winning to effort. If r = 0, the outcome does not depend on effort but depends completely on luck, that is: each contestant has a probability of success of 1/n regardless of what his outlay, xi, is. If r reaches infinity (r → ∞), the probability of winning for a contestant will be completely determined by effort. These kinds of contests are called perfectly discriminating contests. In my thesis however, I will only consider imperfectly discriminating contests (i.e. contests where r is finite). In an imperfectly discriminating contest, an ultimate winner is not designated; but rather, each player is assigned a probability of winning the prize (Nitzan, 1991, p. 1522). When a player i makes a bid for the prize V, his expected profits will be

i i j r j r i r i i i i _{x x} V x x x V p ⋅ − + = − ⋅ = ≠

π

(2.2)

Each player will try to maximise his profits by choosing a suitable xi, while taking into

account that the other players will do the same. This model can been seen as a one-shot game: the profit of player i is its payoff, and the strategy space of player i is simply the possible expenditure he can make. A Nash equilibrium is then a set of expenditures (x1*,x2*,…,xn*) in which each player is choosing it profit-maximising outlay/expenditure given his beliefs about the other player’s choice, and each player’s beliefs about the other player’s choice are actually correct. Baye (1993) has shown that an equilibrium in pure strategies only exists when r ≤ n/(n-1).

2.2.1. Solving the Tullock model

Solving the basic Tullock is therefore done by differentiating equation (2.2) with respect to xi. This yields the first order condition:

Since the basic Tullock contest is a symmetrical game (i.e. all players have the same function defining their chances of winning, and all players compete for the same prize), the outlays, xi, will be the same for all players, therefore (2.3) reduces to:

V x n rx nx rx r r r r ₁ 2 2 1 2 1 = − − − or V x n r x n r 1 1 2 1 ₋ − ₌ −

Solving this for x gives the (equal) expenditure each player will make in the Nash equilibrium: V r n n x_i∗ = − ⋅ ⋅ 2 ) 1 ( _(2.4)

Equation (2.4) indicates that the effort put forward by each player will decrease as the number of participant to the contest increases. This is in accordance with intuition since a larger number of players reduces the chance of winning for each individual player, and with that his incentive to invest. Similarly equation (2.4) indicates that the higher the prize, V, and the higher the marginal return to the expenditures, r, the higher a player’s expenditure will be, which is in accordance with intuition as well.

Earlier in this chapter I described that the rent-seeking model was initially designed to study the welfare losses caused by imperfect market structures, e.g. a monopolistic market. Not only the deadweight loss but also the rent-seeking expenditures by the potential monopolists

should be considered as welfare losses. The ratio between total rent-seeking outlays in

equilibrium and the value of the contested rent, the extent of rent dissipation, is a measure for these welfare losses, and has been used in various empirical applications. The extent of rent dissipation, D, is defined as:

V x D n i= i ∗ = 1 _(2.5)

or, in the Tullock model: n n r

D= ( −1)/ (2.6)

Thus, the initial idea that the total amount of profits obtainable from acquiring a monopoly will be completely dissipated does not have to be true. The rent is totally dissipated when the number of rent-seekers is sufficiently large (and the parameter r is sufficiently large). In small-number contest only part of the rent is dissipated, only a part of the monopoly rents that can be earned, will be a welfare loss.

Numerical Example

that both players will invest 25 (which could be money invested in lobbyist to make the government decide in their favour, or money invested in advertising to get a good reputation, or even money invested in bribery). Both players will have an equal chance of winning (expression (2.2)): p1 = p2 = 0.5, and their profits will both be 25 (the expected total income of

50 minus the initial outlay of 25, equation (2.2)). Since both players invest 25, the total outlay is only half of the prize, the dissipation rate is 0.5, i.e. only 50 can be seen as a welfare loss.

2.2.2. Extended Tullock contests

The Tullock model described above can be extended in a simple manner to include risk aversion, and asymmetry of the players. When identical rent seekers are risk averse, there exists a Vn < V, such that the equilibrium of the rent seeking game with risk-neutral players competing for the prize Vn is the same as the equilibrium of the contest with the risk averse players competing for the prize V (Nitzan, 1994, pp 45-46). Or, Vn is the risk-neutrally equivalent rent which makes each player choose the same outlays in equilibrium, as the risk-averse players would choose in equilibrium with the prize V.

One possible asymmetry of the players could be a different assessments of the value of the opportunity (e.g. both players are companies, but one company values the same rent-opportunity higher because it better fits in his existing activities). Another possible asymmetry is a difference in individual lobbying capabilities, a bias (e.g., one company would be more efficient in his outlays because he already has a good reputation). To incorporate these asymmetries into the model expressions (2.1) and (2.2) are changed into:

≠ + = i j r j j r i i r i i i _{a x a x} x a p (2.7)

where ai, aj > 0 represent the bias of player i respectively player j.

i i i j r j j r i i r i i i i i _{a x a x} V x x a x V p ⋅ − + = − ⋅ = ≠

π

(2.8)

where Vi is the valuation, that player i gives to the prize. The risk-aversion of the players could be taken into account by replacing these valuations of the players for the prize by their risk-neutral equivalents.

2.3. Contest success functions

After having seen that the Tullock model can easily be extended to take into account risk-aversion and asymmetric players, the question arises whether Tullock’s basic equation for success in rent-seeking competitions is also adaptable to different situations. The function that provides each player’s probability of winning as a function of all players’ efforts is called the Contest Success Function (CSF). In the basic Tullock model, described in the previous section, the CSF is given by expression (2.1).

paper, Skaperdas (1996) starts out with five easily interpretable axioms and is able to derive the CSF from these.

The axiomatisations which Skaperdas uses, do not depend on the specifications of the contest or on the particular realisation of efforts. When Skaperdas looks at n-player contests, he defines x = (x1, x2,…,xn) to be a vector of efforts for the n players (Skaperdas, 1996, p. 284). Each player i’s winning probability is denoted by pi(x) (pi: [0,X]n → R where X > xi for all i ∈ N).

Skaperdas uses the following axioms:

(A1) _i∈N = ≥ ∈ > i >

i i

i _{x p x i N x x p x}

p ( ) 1 and ( ) 0for all andall ;if 0 then ( ) 0

(A2) For all i ∈ N pi_{(x) is increasing in x}

i and decreasing in xj for all j ≠ i (A3) For any permutations

π

of N (i.e. a bijection

π

: N → N) we have p i ₍x₎₌ p₍x _,x _,....,x _{n )∀}i_∈N 2 1 ) ( π π π π

The first axiom above (A1) is giving the conditions for a probability distribution function, i.e. all probabilities sum up to one, and probabilities cannot be negative. Axiom (A2) is also a reasonable assumption to make. It is intuitive that the probability of a player winning the contest should increase with his effort, and that his probability of winning should decrease if other player make larger investments. The third axiom (A3) is an anonymity property stating that the probabilities of success do not depend on the identity of the players but only on the efforts they make. This property also implies that players, who put forward the same effort, will have the same chance of winning the contest.

For the remaining two axioms, Skaperdas looks at what assumptions could be reasonable for a group of players, or what the interaction would be between players and a subset of the original set of players. More specific (Skaperdas, 1996, p 285): if a nonempty subset of the players, M ⊆ N, were to break off from the other players and engage in a contest amongst themselves, what would be the probability of success of each player in that subset? Skaperdas defines pmi(x) to be the ith player’s probability of success of a player belonging to the subset M, and states as his fourth axiom:

(A4) p (x) p (x)/ p (x) i M N with at least twoelements

M j j i i m = ∀ ∈ ⊆ ∈

That is, contests among smaller numbers of players are qualitatively similar to those among a larger number of them. Suppose for example that the total set of players, N, consists of three players. Axiom (A4) then states that when a subset M of N, say M ={player 1, player 2}, were to play a contest amongst themselves, with player 3 staying at the sidelines, will give the same results as a contest of only two players N = {player 1, player 2} without subgroups, and without a third player.

while he is competing in the subset M, should not depend on the players not a member of that subset, or:

(A5) pi (x)

m is independent of the efforts of the players not included in the subset (M ⊂

N); or, pi (x)

m can be written as pmi (xm)where xm =(xj; j∈M)

Skaperdas calls the axiom above: ‘the independence from irrelevant alternatives’ property. The last two axioms can be replaced with an axiom stating that the winning probability satisfies Luce’s Choice Axiom (Clark and Riis, 1998, p. 202). That is (A4) and (A5) together can be substituted by:

(A4’) k i x p x p x x x x x pi _{k k n} i _{k ∀ ≠} − = + − ) ( 1 ) ( ) ,.... , 0 , ,..., ( 1, 2 1 1

This is another, but easier, way to state the independence from irrelevant alternatives property. Axiom (A4’) states that the probability that player i wins if player k does not

participate (xk = 0) is equal to the probability that i wins when k participates (xk > 0) given that k does not win. Again this last axiom (or last two, if we look at Skaperdas’ axioms) is not unreasonable: the choice between two alternatives should be independent of an unchosen third alternative.

It can easily be seen that the Tullock CSF, (1), satisfies the first three axioms mentioned above. Maybe the axiom (A4’) is notimmediately evident, however writing

− = = + + − + + = ₁ 1 1 1 1 2 1, ,..., ,0, ,..., ) ( _k j n k j r j r j r i r i n k k i x x x x x x x x x p and = = = = = − ⋅ = − ⋅ = − n j r k r j n j r j n j r j r i n j r j r k n j r j r i k i x x x x x x x x x x p x p 1 1 1 1 1 ) 1 ( 1 ) ( 1 ) (

makes clear that the fourth axiom (A4’) also holds.

The Tullock function belongs to the following class of CSFs that satisfies (A1)-(A5):

∈ = N j j i i _{x f x f x} p ( ) ( )/( ( )) for all i ∈ N (2.9)

After having axiomatised the general form of contest success functions, Skaperdas sets out to axiomatise two special cases of the general additive representation in (2.9). First he defines λx = (λx1,λx2,…,λxn) and then he postulates the following axiom:

(A6) pi₍

_λ

x₎₌ pi₍x _{)for all}

_λ

_{>0andfor all}i_∈N

That is, the contest success function is homogenous of degree zero. Multiplying the efforts of all players by the same factor will not influence the winning probabilities of the players (e.g. if all players were to double their bids, this would have no affect on the outcome of the game). Skaperdas shows that the following functional form satisfies the axioms (A1)-(A6):

∈ = N j r j r i i x x x p

α

α

)

( for some α > 0 and r > 0 (2.10)

Moreover, he shows that (2.10) is the only continuous functional form satisfying all six axioms (Skaperdas, 1996, p 288). Since axiom (A6) shows that the ratio of any two players winning probabilitiesdepends on the ratio of their efforts, I shall refer to the functional from in (2.10) as the ‘ratio-form’ in the remainder of this thesis.

Clark and Riis (1998) show in their paper that by relaxing the third axiom of Skaperdas, the functional form ∈ = N j r j j r i i i x x x p

α

α

) (

can be axiomatised as well. This is the Tullock function with the possibility of a bias, as described in section 2.2.2, and is another example of the ratio form (i.e. the standard Tullock CSF, represented by expressions (2.1) and also the ratio form in (2.10) can be seen as special cases of this function).

The second special case of the general form of contest success functions (2.9) Skaperdas axiomatises is the ‘difference-form’. In contrast to the ‘ratio form’ the probability players in a contest win depends not on the ratio of their efforts, but on the difference between their efforts. Therefore the seventh axiom is:

(A7) pi(x)₌ pi(x₊c) _∀c_∈Rn such that x_{i +}c_≥0,_∀i_∈N ;_{in addition, assume f(.)} is defined for xi = 0

This axiom states that an increase of the vector of efforts with a constant vector with all it components equal to c, will not influence the outcome of the contest. That is, if all the players increase their effort with the same amount (e.g. in a two-player game, player 1 increases his effort from 50 to 55, and player 2 from 60 to 65), the chances of winning will not be affected. The functional form that results from axioms (A1)-(A5) and (A7) is:

2.4. Difference-form contest success functions

In an interesting study, Hirshleifer (1989) applies the rent-seeking model to a situation of military combat, and shows that the ratio-form of the contest success function, as proposed by Tullock, does not always have desirable consequences. He argues that the difference form of the CSF does not have the crucial flaws that he discovered when he used the rent-seeking theory to explain military interactions.

One implication of the Tullock CSF is that a player who invests nothing (has an effort of zero) will certainly lose the contest as long as one of the other players invest a finite amount, no matter how small (see expression (2.1)). In a military situation, Hirshleifer argues, this may not be a desirable consequence, because one country may surrender rather than resist and fight his opponent. It might make sense to surrender to an aggressor, if the submitting nation does not expect to lose absolutely everything by giving up the struggle. This is reasonable, since in general it will be costly, even in the absence of resistance, to locate and extract all the possible recourses from the submitting country (Hirshleifer, 1989, p 103).

Another implication of the ratio model would be that peace, when defined as a situation where both players do not invest (i.e. x1 = x2 = 0), can never be a Nash equilibrium. Looking at expressions (2.1) and (2.2), it is clear that if the players were in a situation where neither invested, both players would have an incentive to defect. As long as V is larger than zero, a player could increase his probability of winning to 100% by investing only a very small amount. Therefore, a long-lasting peace could never be sustainable in the traditional ratio model.

A third and final implication of the traditional model is that the inflection point (i.e. the point where increasing returns to effort change into diminishing returns) of the CSF cannot lie at the point where both players invest the same amount. And Hirshleifer argues, that in military struggles there is an enormous gain when your side’s forces gain from just a little smaller than the enemy’s to just a little larger (Hirshleifer, 1989, p.103). To see this, the contest success function of player 1, in a two-player contest has been drawn for different values of the parameter r (while keeping the effort of player 2 constant at 100) in figure 2.1.

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 50 100 150 200 250 300 x1 p1 r=0.5 r=1 r=3

Figure 2.1: Contest success functions in the ratio-form

but after the inflection point (at x1≈ 80, when r = 3), the CSF shows decreasing returns to effort. Hirshleifer proves that the inflection point always lies to the left of the point where both players invest the same amount (the point x1 = x2 =100 in figure 2.1). Therefore, it will not be possible that the ratio-form CSF has increasing returns to effort at the point where both players invest the same amount. Alternatively, in the military context, it will not be possible that there exist increasing returns to scale at the point where both armies are of the same size. However, when the difference-form of the contest is used the three undesirable consequences described above do not occur. For a two-player contest the CSF for player 1 can be written as:

)} ( exp{ 1 1 1 2 1 _{k x x} p − + = (2.12)

where p2 is defined accordingly, and where k (with k > 0) is the marginal return to expenditures. Note that this is the two-player version of expression (2.11).

As can be seen from (2.12) one-sided surrender is possible when the difference-form is used. (Although Hirshleifer uses military examples to explain the drawbacks of Tullock’s original CSF, the undesirable implication of that CSF could of course also occur in other, non-military, applications of the rent-seeking model). Substituting the value x1 = 0 into (2.12), does not imply that player 1 will lose with certainty (p1 does not become zero).

In addition, a situation of two-side peace is possible when expression (2.12) is used. The expected profits of the players are:

i n j j i i i i V x kx kx x V p ⋅ − = ⋅ − = =1 ) exp( ) exp(

π

(2.13)

Setting x2 = 0 in the two player contest, and taking the derivative of (2.13) for player 1, leads to the first order condition:

V kx kx k 1 )) exp( 1 ( ) exp( 2 1 1 ₌ − + −

A similar first order condition can be derived for player 2 (the players are completely

symmetrical). For x1 = 0 to be a solution, V has to equal 4/k, therefore, when for example k = 0.04, V = 100, the efforts x1 = x2 = 0 can be an equilibrium solution of the model.

0 0,2 0,4 0,6 0,8 1 1,2 0 50 100 150 200 250 300 x1 p1 k=0 k=0.02 k=0.04

Figure 2.2: Contest success functions in the difference-form

Although the difference-form, as shown in this section, is very useful in modelling certain types of rent-seeking contests, there are also some disadvantages to using this form. The difference-form, implies that the winning probabilities of, say, x1 = 1 and x2 = 2 are the same as those of x1 = 10001 and x2 = 10002 (Skaperdas, 1996, p 289). Another disadvantage is that, as proven by Baik’s study (1998), if a pure-strategy equilibrium exists, it is unique and only one of the players expends effort or neither player expends effort. A pure-strategy equilibrium in which both players invest appears not to be possible.

2.5. A ratio-form contest success function with a noise parameter

From the previous section, it has become clear that both the Tullock and the difference-from contest success function have some flaws. In a recent paper, Amegashie (2006) suggests a CSF that appears not to have these flaws. He proposes a simple variant of the Tullock

function, which has the latter function as a special case under certain conditions, but does not have its undesirable characteristics. Since Amegashie’s function is of the ratio-form it will not have the undesirable features of the difference-from CSFs either.

Amegashie proposes the following contest success function:

≠ + + + = i j j i i i _{x x n} x x p

α

α

) ( (2.14)

where x = (x1, x2,…,xn) is again the vector of efforts of the players, n (with n ≥ 2) is the number of player competing in the contest, and

α

(with

α

≥ 0) is a parameter that captures the degree of noise. If

α

= 0, then Amegashie’s contest success function gives the Tullock function with r = 1 (see expression (2.1)).

In the basic Tullock model (as described in section 2.2), the parameter r reflects the marginal return to the lobbying expenditures. The higher is r, the more sensitive is the probability of winning the tournament to effort. If r = 0, then each player has a probability of success of 1/n regardless of his effort. That is, the parameter r captures the degree of noise in the Tullock model. In Amegashie’s contest success function, the effectiveness of the lobbying

expenditures of the players will have no influence (this corresponds to the case where r = 0, in the Tullock model).

An important feature of this CSF is that the i-th player’s probability of success is positive, even if he expend zero effort, and some player expends positive effort. In the Tullock model, a player investing nothing while another player does invest (no matter how small his

investment) has no chance of winning. This is one of the reasons why Hirshleifer found the Tullock model unsuited for modelling military combat. Fortunately, the new CSF relaxes this feature of the Tullock model. Assuming

α

> 0, there is noise at every possible combination of efforts. This is in contrast with the Tullock model where, no matter what the value of r is, there is no noise if one player invests nothing, and another player does invest.

In the remainder of this chapter, I will first discuss in more detail whether Amegashie’s CSF satisfies the simple axioms of Skaperdas. Then I will solve then model, and finally, I will show that the solution of Amegashie’s model does not have the same undesirable features as the Tullock model and models based on the difference-form CSF.

2.5.1. Properties of the CSF with a noise parameter

The first of the axioms proposed by Skaperdas (see section 2.3) states that the CSF satisfies the conditions of a probability function. It can easily be seen from the expression (2.14) for Amegashie’s CSF that pi(x) ≥ 0 for all i ∈ N and all possible vectors x. Also the probabilities sum up to one, i.e. n₌ =

i 1pi(x) 1.

That the CSF with a noise parameter satisfies the second axiom is also obvious. The winning probability is increasing in player i’s own effort, in xi, and it is decreasing in any other player’s effort, in xj for any j.

Skaperdas’ third axiom also holds. Axiom (A3) is the anonymity axiom, stating that each player’s probability of winning should not depend on his identity. From (2.14), it can be seen that changing a player’s identity will not affect the outcomes of the contest. That is changing i to for instance k, will result in a new winning probability pk which is equal to the original probability pi, as long as the efforts of player i stays the same when given the identity k. That (A3) implies that all players who invest exactly the same amount will have the same

probability of winning can also seen to be true in Amegashie’s new CSF.

Although the new CSF does satisfy the anonymity axiom (A3) of Skaperdas, it can be shown (Clark and Riis, p. 202) that this axiom is not a necessary condition to axiomatise a CSF. This is important because later in this thesis I will introduce a bias-parameter to Amegashie’s function. The introduction of a bias parameter will account for different treatment of the contestants (e.g. due to affirmative action programs one player might get the same results as other players with a considerable lower investment). Therefore, when a bias is possible, the probability of success is no longer dependent only upon the players’ investments, the anonymity axiom does not hold anymore, but the axiomatisation is still valid.

Following Clark and Riis, I replace Skaperdas’ fourth and fifth axiom by Luce’s Choice Theorem, by: (A4’) k i x p x p x x x x x p k i n k k i − + =_{1− ( )} ∀ ≠ ) ( ) ,.... , 0 , ,..., ( _{1, 2 1 1}

that the new CSF satisfies this axiom, I first substitute (2.14) in the left part of (A4’) to get the chance that player i wins when k does not enter the contest:

≠ + − _{+ −} + = k j i i n k k i _{x n} x x x x x x p

α

α

) 1 ( ) ,... , 0 , ,..., , ( 1 2 1 1 (2.15)

Substituting (2.14) in the right part of (A4’) expresses the chance that player i wins, when player k does participate, given that he does not win:

≠ ≠ _{+ +} + − ⋅ + + + = − k j j k k i j j i i k i n x x x n x x x x p x p

α

α

α

α

1 1 ) ( 1 ) ( or = = = = + + − + + ⋅ + + = − n j j k n j j n j j n j j i k i n x x n x n x n x x x p x p 1 1 1 1 1 ) ( 1 ) (

α

α

α

α

α

α

or ≠ = = = + − + = − − + + ⋅ + + = − n _{k j k j}i j j n j j n j j i k i n x x x n x n x n x x x p x p

α

α

α

α

α

α

α

) 1 ( ) ( 1 ) ( 1 1 1 (2.16)

Since (2.15) and (2.16) are equal the contest success function of Amegashie satisfies axiom (A4’). The winning probability of player i is independent of irrelevant alternatives.

Therefore, the CSF with a noise parameter satisfies all five intuitively reasonable axioms proposed by Skaperdas. This means that Amegashie’s CSF belongs to the class of CSFs as given by expression (2.9) in section 2.3. It belongs to the class of CSFs that was proven by Skaperdas to be the only representation possible for a CSF (assuming that it satisfied five reasonable and intuitive axioms).

However, the CSF with a noise parameter does not belong to one of the two special cases axiomatised by Skaperdas. In contrast to the Tullock function, it is not homogenous of degree zero (axiom (A6)), and it does not satisfy axiom (A7) either, i.e. it is not of the difference-form either.

To show that the new CSF is not homogenous of any order I will look at a two-player contest. Suppose that player 1 makes a larger investment than player 2, i.e. x1 > x2. This gives a

winning probability for player of:

α

α

2 ) , ( 2 1 1 2 1 1 _{+ +} + = x x x x x p (2.17)

CSF would be homogenous of degree zero, like the Tullock function, the winning probability would not change (i.e. p1(x1,x2) = p1(

λ

x1,

λ

x2)). Substituting

λ

x1 and

λ

x2 in (2.17) gives:

α

λ

λ

α

λ

λ

λ

2 ) , ( 2 1 1 2 1 1 _{+ +} + = x x x x x p (2.18)

Subtracting expression (2.17) from expression (2.18) leads to:

) 2 )( 2 ( ) 2 )( ( ) 2 )( ( ) , ( ) , ( 2 1 2 1 2 1 1 2 1 1 2 1 1 2 1 1

_λ

_λ

_α

_α

α

λ

λ

α

α

α

λ

λ

λ

+ + + + + + + − + + + = − x x x x x x x x x x x x p x x p or ) 2 )( 2 ( ) )( 1 ( ) , ( ) , ( 2 1 2 1 2 1 2 1 1 2 1 1

_λ

_λ

_α

_α

λ

α

λ

λ

+ + + + − − = − x x x x x x x x p x x p (2.19)

Since I assumed that

λ

> 1, and that x1 > x2, it follows from (2.19) that when

α

> 0, the expression (2.19) is greater than zero (

α

= 0 is the special case of the new CSF which gives the Tullock function). Therefore the probability of winning for player 1, is not the same after a proportional increase in the efforts of both players, and thus the new CSF is not

homogenous of degree zero. Moreover, it is not homogenous of any degree.

This may be desirable in certain situations, since although the players increased their efforts with the same proportion, the absolute increase of player 1’s effort is bigger than the absolute increase of player 2’s effort. That is:

λ

x1-x1 is bigger than

λ

x2-x2, (because of the assumption x1 > x2). A bigger absolute increase in player 1’s investment relative to player 2’s investment might, in certain situations, demand an increase in the chance that player 1 wins and a

decrease in player 2’s winning probability. When the CSF is homogenous of degree zero this is not possible.

To show that the CSF as given in (2.14) is not of the difference form either, I will again look at a two player contest. In addition, I will again assume that player 1 puts forward a greater effort than player 2, i.e. x1 > x2. For a CSF to be of the difference form, it has to satisfy axiom (A7), which states that the winning probability for a player will not change when every player increases his investment with the same amount. Let c be the amount with which every player increases his effort. Then the probability of winning for player 1 is:

α

α

2 ) , ( 2 1 1 2 1 1 _{+ + + +} + + = + + c x c x c x c x c x p (2.20)

Subtracting the winning probability before the increase (which is again given by expression (2.17)) form the winning probability after the increase, leads to:

An increase of the efforts of both players with the same amount, c, results in a decrease of the winning probability of the player with biggest initial outlay. Therefore, the CSF does not satisfy assumption (A7), and it is not of the difference form.

Summarising, Amegashie’s CSF does satisfy the first five axioms proposed by Skaperdas, and it belongs to the general category of possible CSFs. However, it does not satisfy axiom (A6) or axiom (A7), and it does not belong to one of the special cases axiomatised by Skaperdas. Amegashie has not been able to axiomatise his function. He has not been able to propose an axiom (like the axiom (A6) or (A7)), which his function would satisfy, and then prove that his function is the only function that satisfies that axiom. However, he argues (Amegashie p. 138), that his success function is more tractable than Tullock’s function in certain cases, and that axiomatisations by themselves are unlikely to settle the issue of appropriateness of a CSF for any particular situation. Amegashie states that he hopes his contest success function will be judged on these grounds.

Unfortunately, I have not been able to axiomatise Amegashie’s CSF either. However, in the chapters 3 and 4, I will examine the appropriateness of his function in a few situations. 2.5.2. Solving Amegashie’s model

In section 2.2.1 I have given the solution for the Tullock model for a contest with n identical players with a varying mass effect or noise parameter r. Although that solution was easy to derive, it becomes more difficult to derive a solution when the players are not identical. When the players have a different valuation of the prize (the addition of a bias parameter will be examined in the next chapter), it becomes very difficult, if not impossible to solve the Tullock model with a varying noise parameter. Amegashie (2006, p. 138) notes that in the literature on rent seeking, only general solutions have been calculated for a n-player contest with r = 1, or for a two-player contest in case r ≠ 1.

Therefore, if a model based on Amegashie’s CSF can be solved, it will be possible to incorporate noise in an n-player contest with non-identical players, which appears to be impossible in the Tullock model.

Assume that there are n contestants competing for of prize of value V, where the ith contestant has a valuation of this prize of Vi. The expected payoff of player i is:

i i i j j i i i i i i _{x x n} V x x x V p ⋅ − + + + = − ⋅ = ≠

α

α

π

(2.22)

Similar to the payoff function I used to solve the Tullock model in section 2.2.1, the payoff is the difference between the expected revenues and the cost (and these cost are simply given by the investment xi of player i). In contrast to the payoff function used in section 2.2.1, it is possible that players value the prize differently. In addition, the probability that a player wins that prize is now defined by Amegashie’s CSF (expression (2.14)), instead of being defined by the Tullock function.

(

)

(

)

(

1

)

₂ 1 1 1 1 _{⋅ −} + ⋅ + − ⋅ + = ∂ ∂ = = i n j j i n j j i i _V n x x n x x _α α α π

Thus, the first order conditions are:

(

X n

)

V i x n X i i _{⋅ − = ∀} + + − + 1 0 1 2

α

α

α

(2.23) where X ≡ n _i i=1x .

Summing (2.23) over all n contestants gives:

(

)

= = ₌ + + − + n i i n i i V n X n x n X n 1 2 1 1

α

α

α

or = = + − + n i i V n X n X n 1 1 1 α α

and from this, the equilibrium aggregate effort (i.e. the sum of the efforts all the players will make in the equilibrium) can be calculated to be:

α n V n X n i i − − = = ∗ 1 1 1 (2.24) After defining = − ≡ n i i V n V 1 1

1 _{, the equilibrium aggregate effort (2.24) can be substituted back}

into (2.23) to give:

(

)

i i i V V x V n n V x n n V 1 1 1 2 2 = + − = + − + − + − α α α α α α

and solving this for xi results in the following equilibrium effort for player i:

α

α

= − − − − = ∗ i i i _V V V V V V V x 2 2 1 (2.25)

If there is no noise (α = 0), the efforts put forward by the players will be the highest (and will be the same as the efforts in a Tullock contest with r = 1). Further, the investment made by player i will by greater than the investment made by player j if the valuation of the prize by player i is greater than the valuation of player j. Subsequently the winning possibility of player i in equilibrium will also be larger than that of player j (a larger effort substituted in (2.14) results in a greater chance of winning), if contestant i’s valuation is greater than that of contestant j. This is in accordance with intuition as well, since the larger (the valuation of) the prize, the more a player will invest to win it, and the more a player invest, the higher the probability that he wins it.

To compare the results derived above with the results from the Tullock function, I look at the case where Vi = V for all i. That is, all players are identical. In this case, (2.24), the aggregate effort of all contestants, changes into:

α n V n n X∗ = ( −1)⋅ − _(2.26)

The effort of player i in equilibrium can be derived by dividing the above expression by the number of players, n (since all player are equal, there will be a symmetric equilibrium, and all player’s will invest the same amount).

α − ⋅ − = ∗ _V n n x ( ₂1) (2.27)

Note that for the effort of the players to be positive (i.e. x* > 0), the valuation of the prize has to be sufficiently high, or the amount of noise, α, has to be sufficiently small.

The efforts in the Tullock contest (with r = 1) are the same as the efforts in Amegashie’s model with α = 0, as might be expected (see expression (2.4) in section 2.2.1). In Tullock, an increase in the number of players always results in an increase in aggregate efforts, and

likewise in a higher dissipation rate. This is not necessarily the case in Amegashie’s model. The derivative of (2.26) with respect to n is given by:

α − = ∂ ∂ ∗ 2 n V n X

Hence, an increase in the number of contestants will result in an increase in aggregate efforts (and consequently in an increase in the dissipation rate), if and only if, the players’ valuation of the prize is sufficiently large. In most applications, the value of the prize will be

considerably larger than the amount of noise; nevertheless, in theory it is possible that the dissipation rate will decrease as a result of an increase in the number of contestants.

Above it has been shown that, in contrast with the Tullock model, noise can be incorporated in Amegashie’s model when the players are not identical. Now I will address the undesirable features of the Tullock model mentioned by Hirshleifer. As I described in section 2.4,

derivative of the payoff function with respect to player i’s effort to be smaller or equal to zero, that is: i V n X x n X x i i i i _{⋅ − ≤ ∀} + + − + = ∂ ∂ 0 1 ) ( 1 2

α

α

α

π

(2.28)

When (2.28) holds, an increase in effort for every player will result in a decrease in his payoff. Consequently, no player will have an incentive to change his investment, and thus the situation where no one invests will be an equilibrium.

Substituting xi = 0 ∀ i in (2.28) gives: i V n n n V n n V n n − ⋅ i − = − ⋅ i − = ⋅ − ⋅ i −1≤0 ∀ 1 1 1 1 1 1 ) ( 1 2 2 2 2

_α

_α

_α

α

α

α

or i V n n i ≤ ∀ ⋅ − ) 1 ( 2 α (2.29)

When the condition (2.29) holds an equilibrium where no one invests is a possible outcome of Amegashie’s model. That is, when the level of noise is sufficiently high it will be that, in terms of military action, peace can be an equilibrium outcome of the contest.

2.6 Summary

This chapter has set out the basic elements of a rent-seeking model. Since the rent-seeking theory started with Tullock, and since many studies in the literature on contests use his model, I described and solved the basic Tullock model at the beginning of the chapter.

Skaperdas showed that the contest success function used by Tullock is only one of the few possible CSFs that can be used. Starting out with five simple and reasonable axioms, Skaperdas has proven that there is only one possible class of CSFs. In addition, Skaperdas axiomatised two functional forms within this class of CSFs: one in which the winning

probabilities in the contest depend on the ratio of the efforts of the players, and another one in which the winning probabilities depend on the difference in efforts. The former functional form, to which the Tullock function belongs, is called the ratio-from, and the latter the difference form.

Hirshleifer used the application of the rent-seeking model to military combat to show that the Tullock function has some undesirable features. With the Tullock CSF, it is not possible to model one-sided surrender or peace, i.e. it not possible to have a Nash equilibrium in the Tullock setting where one player or both players expend zero effort.

In the last section of this chapter, I described a contest success function that does not have the undesirable features that the Tullock function, and functions of the difference-form have. This function was proposed by Amegashie, and has the Tullock function as a special case under certain conditions. Amegashie’s CSF does make a Nash equilibrium where one or both players do not invest possible, and since it is not of the difference form it does not have the undesirable features of Hirshleifer’s function either.

3. Stackelberg rent seeking with the new contest success function

In the previous chapter, I discussed the contest success function proposed by Amegashie. According to Amegashie the new contest success function will be more tractable than the Tullock CSF, and it will show the degree to which luck as opposed to effort affects behaviour in contests. In this chapter, I will use a well-known study in the rent-seeking literature to test these claims. Although Amegashie himself already studied the effects of his new CSF in the standard Tullock setting, papers where the nature of the contest is different have not yet been studied. Instead of considering a rent-seeking contest modelled as a static non-cooperative game, I will consider a game of a more dynamic nature. In particular I will study the

Stackelberg rent-seeking model of Leininger (1993) which is an interesting example showing how the static Tullock game can be extended into a dynamic model. By changing the CSF in Leininger’s paper into Amegashie’s CSF, I will examine what the consequences of noise are on Leininger’s model. Moreover, I will examine whether the Amegashie’s new CSF is tractable and applicable in modelling dynamic rent-seeking contests.

This chapter is organised as follows: in section 3.1 the paper written by Leininger will be described, in section 3.2 the noise parameter will be added to Leininger’s model, and the effects of this new CSF on the outcomes presented in Leininger’s paper will be discussed.

3.1. The standard Stackelberg contest

Leininger noticed that the standard Tullock contest could lead to the strange phenomena that players make irrational high bids, i.e. bids that are higher than the expected value of their payoffs (Leininger, 1993, pp 56-57). Suppose for example that the prize in a two-player contest has a value of 100, and that the marginal returns to effort are increasing, say r = 3. Then according to Tullock (see expression (2.4) in section 2.2), each player will bid 75 for an expected revenue of 50. If a player bids 75 for an expected revenue of 50, his expected profit

πi* would become negative, and it would be individually rational not to bid at all. However, if the other player also acts individually rational, and does not bid, it becomes profitable again for the first player to bid (a very small bid would win him the prize if he knew the other the other player would not enter the contest). Thus, the players might be drawn in a wasteful competition, which would escalate the bids up to 75. This phenomena of players making irrational high bids is caused by the non-existence of a Nash-equilibrium for r > n/(n-1) (see also section 2.2).

3.1.1. Leininger’s basic model

The basic model consists of a two-player, two-stage contest in which a player’s strategy is made up out of two components: the timing of his move and the size of his outlay. Leininger considers two possible timing decisions, either a player moves ‘Early’ or a player moves ‘Late’. When both players decide to move early or both decide to move late a simultaneous game results, and the next stage will be a standard Tullock game. When the timing-decisions of the players are different, that is one player chooses early and the other chooses late (or vice versa), a sequential game arises, the second stage will be a Stackelberg model.

Leininger uses the words ‘Early’ and ‘Late’ and not ‘Earlier’ and ‘Later’ because the players cannot choose if they want to be leaders or followers as this always depends on the timing-decision of the other player. A player decides on his timing without knowing what the other player will choose, after they both made their decision the results are revealed and both players will know what the order of play will be (and will have to play in that order, i.e. the timing decisions are assumed to be binding). That is, the extended game has the following two-stage structure (Leininger, p. 46):

Stage I: Players simultaneously decide whether they move ‘Early’ (E) or ‘Late’ (L). Stage II: The result of stage I is revealed and the rent-seeking game is played with

simultaneous moves, if the result is (E,E) or (L,L), and sequential moves, if the result is (E,L) or (L,E). (In the latter case player 1 is the leader after (E,L) and player 2 is the leader after (L,E)).

In figure 3.1, this game is depicted as a stylised game tree. Both players decide

simultaneously on their timing-decision, is modelled in this figure as first player 1 deciding if he wants to move early or late, and second player 2 choosing between early and late. Since they move simultaneously, player 2 does not know what choice player 1 has made, he cannot distinguish between nodes g1 and g2. After both players made their timing-decision, it will be revealed to them in which of the information sets g3-g6 has been reached. Therefore, they will know what type of game will be played at Stage II.

The decisions made in Stage I will depend on the expected outcomes of both players at Stage II. That is, as well player 1 as player 2 will base their decision of moving early or late on the payoffs of the four possible games that can be played in Stage II (the four possible

information sets g3-g6).

Therefore, Leininger uses backward induction to solve the extended game. First the four possible subgames in Stage II are solved, and after that the equilibrium expected payoffs of these four subgames are used in the resulting reduced game to solve for the timing decisions of both players (to solve Stage I). Since the timing-decision (E,E) leads to the same

simultaneous game as the timing decision (L,L), only three possible subgames have to be solved: the simultaneous game, the game with player 1 leading and player 2 following (E,L), and the game with player 2 leading and player 1 following (L,E).

3.1.2. Simultaneous play

When both players in Stage I have made the same timing-decision ((E,E) or (L,L)), the subgame that is played, is the standard Tullock game. I will use the same notation as Leininger, and denote player 1 as player X, player 2 as player Y, and their outlays as x (respectively y). Both players compete for a rent (prize) of size (value) V. It is possible that the players value the prize differently, Vx is the valuation of the prize by player X, and Vy is

the valuation of the prize by player Y. The probability for player X to win when he makes a bid (outlay) of x is:

y x a x a y x px _{⋅ +} ⋅ = ) , ( (3.1)

where a is a positive bias parameter, which is a measure for the efficiency of player X’s outlay. And the probability that player Y wins when he makes a bid of y is:

y x a y y x py( , )= _{⋅ +} (3.2)

Consequently, when player X makes a bid of size x for the prize his expected profit will be: x V y x p y x x x x( , )= ( , )⋅ − π (3.3)

Similarly the expected pay-off for player Y will be: y V y x p y x y y y( , )= ( , )⋅ − π (3.4)

x V v a v a x ⋅ + ⋅ = ₂ ) ( * (3.5)

where v = Vy/Vx. And for player Y:

y V v a v a y ⋅ + ⋅ = ₂ ) ( * (3.6)

Leininger notes that with different valuations the players will submit different bids. In contrast, with identical valuations – irrespective of the bias – they will submit identical bids. (Leininger, p.48). The accompanying equilibrium profits are given by:

⋅ + ⋅ + = ∗ ∗ y x y x V v a v V v a a 2 2 2 2 ) ( , ) ( ) , (

π

π

(3.7)

From (3.5) trough (3.7) it can be seen that the higher a player’s valuation of the prize is, the higher his outlay to win the prize will be, and the higher the expected pay-off in equilibrium will be. The same applies to the bias: the higher the bias in favour of player X (a > 1), the higher his outlay and expected profit, a bias in favour of player Y (a < 1), means a lower profit for player X.

3.1.3. Sequential play with player X leading

When player X has chosen ‘Early’ in Stage I, and player Y has chosen ‘Late’ (E,L), the game played at Stage II will be a Stackelberg model with player X the leader and player Y the follower, instead of the Tullock game described in the previous section.

Since player X moves first, player Y can observe player X’s bid, x, and choose his own optimal bid given this x. Player Y’s problem is therefore straightforward: given x, player Y wants to maximise his profits:

y V y x a y y x _y y( , )= _{⋅ +} ⋅ − π (3.8)

The first-order condition for this problem is simply:

1 ) ( _{⋅ +} 2 ⋅ = ⋅ y V y x a x a (3.9)

⋅ ≥ ⋅ ≤ ⋅ − ⋅ ⋅ = y y y V a x V a x x a V v a x y ₁ if 0 1 if ) ( ) ( 2 / 1 (3.10)

The above expression is non-negative for values of x between x = 0 and x = 1/a*Vy, after

which y(x) remains zero, which means the follower will only bid, up to a point where the opening bid of the leader is so large that Y will not even enter the contest.

Player X, the leader will look ahead, and anticipate the behaviour of player Y when he decides on his opening bid. He knows that the bid player Y makes will be on the line y(x). Thus, player X wants to maximise:

x V x y x a x a y x _x x _{⋅ +} ⋅ − ⋅ = ) ( ) , ( π (3.11)

or, substituting y(x) in (3.11):

⋅ ≥ − ⋅ ≤ − ⋅ ⋅ = y x y x y x V a x x V V a x x V V x a y x 1 if 1 if ) , ( 2 / 1

π

(3.12)

This function is depicted in the graph below (with the following parameters: Vx = 100, Vy =

120 and a = 2). Left of the intersection of both lines in figure 3.2, the function (3.12) follows the curved line. After the intersection (at x = 1/a*Vy), the opening bid of player X is large

enough to keep player Y from competing, and (3.12) is drawn by the downward sloping straight line.

Profit player X - outlay X (Leininger)

-80 -60 -40 -20 0 20 40 60 80 100 120 0 20 40 60 80 100 120 140 160 180 Outlay player X (Xl) P ro fit p la ye r X (P Ix )

Profit player X, both competing Profit player X, player Y not competing Maximum

In order to determine the optimum bid of player X for x ≤ 1/a*Vy, for the case when both

players offer a positive bid, first the first-order condition has to be determined:

1 2 1 ⋅ 1/2 _{⋅ ⋅ =} x y y V V a V x a (3.13)

which results in an optimal outlay for player X of:

x y x _V v a V V a x = ⋅ ⋅ ⋅ = ∗ 4 4 2 (3.14)

When x ≥ 1/a*Vy, as can be seen in figure 3.2, the profit is decreasing in x (player X will

certainly win the prize, since Y does not compete, therefore an increase in his outlay will only decrease his profit), thus in this case (3.12) reaches it’s maximum at x = 1/a*Vx.

Since x ≤ 1/a*Vx if and only if Vx≤ 2/a*Vy, player X’s optimal bid can now be specified as:

≤ ⋅ ⋅ ⋅ ≤ ≤ ⋅ = ∗ 2 if 1 2 0 if 4 x y y y x x L V V a V a V a V V v a x (3.15) and, substituting (3.14) in (3.10): ≤ ⋅ − ⋅ ⋅ ≤ ≤ = ∗ 2 if 2 1 2 2 0 if 0 y x x x y F V V a v a V a V a V y (3.16)

And finally substituting the equilibrium bids of both the players back into (3.11) and (3.8) will give the outcome of this subgame in case (E,L) has been chosen in Stage I:

⋅ ≤ ⋅ − ⋅ ⋅ ≥ ⋅ − = y y x y y x F y L x V a V v a V v a V a V a V 2 V if 2 1 , 4 2 V if 0 , 1 ) , ( x 2 x

π

π

(3.17)

3.1.4. Sequential play with player Y leading