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Delegation with multiple instruments in a

rent-seeking contest

Lambert Schoonbeek∗

Department of Economics

University of Groningen

The Netherlands

Abstract We consider delegation in a rent-seeking contest with two play-

ers, where delegates have more instruments at their disposal than the main

players. We endogenize both the decision to hire a delegate and the contin-

gent fee offered to the delegates. We characterize the situations when either

no, one or two players hire a delegate in equilibrium. We show that the

decision to hire a delegate depends in a non-monotone way on the size of

the contested prize.

Keywords: Contest, delegation, multiple instruments.

JEL classification code: D72.

∗L. Schoonbeek, Department of Economics, University of Groningen, P.O. Box

800, 9700 AV Groningen, The Netherlands.

L.Schoonbeek@rug.nl. I thank Peter Kooreman, Jos´ e-Luis Moraga-Gonz´ alez and Barbara

Winkel for helpful discussions.

Tel.:+31 50 363 3798. E-mail:

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1Introduction

This paper investigates endogenous delegation in the well-known rent-seeking

contest of Tullock (1980). We recall that in the standard Tullock contest

two players compete for a single prize. Each player exerts effort in order to

increase the probability that he wins the prize. Tullock’s model has spawned

a vast literature and has been applied in many areas, like lobbying, environ-

mental regulation, litigation and sporting; see e.g. Nitzan (1994), W¨ arneryd

(2000), Liston-Heyes (2001), Lockard and Tullock (2001) and Szymanski

(2003). We consider here an extension of a two-player contest in which each

(main) player has the option to either compete himself or to hire a delegate

who competes on his behalf. Examples with delegation abound: firms can

hire professional lobbyists to acquire monopoly rents from the government,

or firms can hire lawyers to win lawsuits, etcetera.

Intuitively speaking, a reason why a player might decide to hire a dele-

gate could be that the delegate in some sense has a larger proficiency than the

player himself. We formalize this intuition by considering a model where a

delegate has the option to compete with two instruments, whereas a player

himself can use only one instrument. An example could be a case where

a firm wants the government to change the law such that it acquires a

monopoly rent. It might be that then the firm can only influence the gov-

ernment’s decision by lobbying political parties in the parliament, while a

professional lobbyist, in addition to this, might also put pressure on the gov-

ernment’s decision by e.g. using her (direct) contacts with influential officials

within the government or by advancing litigation for new precedents. The

specification of our model borrows from Epstein and Hefeker (2003), who

presented an extension of the standard two-player Tullock contest (without

delegation) in which the players can compete with two instruments rather

than with just one. Epstein and Hefeker show that the results derived with

their model significantly differ from those derived with the standard Tul-

lock contest. Hence, one cannot innocently replace two instruments by one

instrument, effort.

We assume that a player cannot observe the efforts exerted by his dele-

gate. In order to cope with the moral hazard, each player offers his delegate

a contingent fee, i.e. a delegate obtains a deliberately chosen fee if she

wins the contest and nothing otherwise. We investigate the following non-

cooperative three-stage game. In stage 1, each player decides whether or not

to hire a delegate. If a player decides to hire a delegate, then in stage 2 he

selects the delegate’s contingent fee. In stage 3 we have the actual contest

for the prize. If a player has not hired a delegate, then he competes himself

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in this stage; otherwise, his delegate competes on his behalf.

We derive the conditions under which no, one or both players decide to

hire a delegate in a (subgame-perfect pure-strategy) Nash equilibrium and

give the corresponding contingent fees. We establish the interesting finding

that the decision to hire a delegate does not depend in a monotone way

on the size of the contested prize. This is related to the fact that in some

equilibria delegates optimally use only one instrument, whereas in others

they prefer to use two instruments.

Some other papers have studied delegation (where delegates have only

one instrument at their disposal) in two-player contests. Baik and Kim

(1997) assume that delegates have a greater so-called ability than the players.

This means that, ceteris paribus, if a delegate exerts a certain effort, then

this has a larger positive effect on the probability that the player associated

to her will win the prize than if this player exerts the same effort level

himself. Baik and Kim also endogenize the decision to hire a delegate – they

show that if a player hires a delegate, then the ability of this delegate must

exceed his own. However, in their analysis the payment schemes offered to

the delegates are exogenously given. They assume that a delegate receives

an exogenously given contingent fee, and in addition to that a fixed fee

(which depends on her ability) regardless of the outcome of the contest.

W¨ arneryd (2000) investigates a two-player contest where it is exoge-

nously given that both players must hire a delegate. He also assumes that

the delegates and players have identical abilities. For this situation, he en-

dogenously determines the equilibrium size of the contingent fee. In fact, it

turns out from his analysis that if the players would be able to decide to hire

a delegate or not, then they would face a prisoners’ dilemma: i.e. not hiring

a delegate would be a dominant strategy for each player, but both would

benefit if both would hire a delegate. Delegation does not endogenously

arise in equilibrium in such a case, however.1We stress that we endogenize

both the decision to hire a delegate and the contingent fee. Moreover, we

identify equilibria in which delegation does occur. We finally mention that

Schoonbeek (2002) examines one-sided endogenous delegation in a contest

where the two players have different risk-attitudes, while Schoonbeek (2004)

analyses endogenous delegation in a contest between two groups of players.

The paper is further organized as follows. In Section 2 we present the

model and derive the equilibria. We conclude in Section 3. All proofs are

in the Appendix.

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2The model and the equilibria

Consider a contest with two risk-neutral players i = 1,2, in which one of the

players can win a prize of value V > 0. The contest is modelled as a non-

cooperative three-stage game. In stage 1, each player decides whether or not

he hires a risk-neutral delegate who will compete on his behalf in stage 3. If

a player does not hire a delegate, he will compete himself in stage 3. In stage

3, the relevant contestants compete for the prize by exerting nonrefundable

and nonnegative efforts. The effort of player i (if relevant) is denoted as ei

(i = 1,2). A delegate, if hired, can compete with two instruments in stage 3.

The effort levels of these two instruments chosen by delegate i (if relevant)

are denoted as yiand zi(i = 1,2). Depending on the players’ decisions in

stage 1, we distinguish four possible cases in the remainder of the game: in

case (a) both players do not hire a delegate, in case (b) only player 1 hires

a delegate, in case (c) only player 2 hires a delegate, and in case (d) both

players hire a delegate. If player i decides to hire a delegate, then this player

offers delegate i in stage 2 a contract that specifies her payment. There is

moral hazard since player i cannot observe the effort of delegate i in stage

3. Delegate i is offered a contingent fee contract; she receives a fraction

0 ≤ γi ≤ 1 of the prize V if the prize is won for player i, and nothing

otherwise. Player i selects the value of γi. Delegate i accepts the contract

if her expected payoff is nonnegative.

The probabilities that player 1 and player 2 receive the prize are de-

noted by, respectively, q and 1−q. The probabilities depend on the relative

magnitudes of the efforts of the actual contestants. We use the specification

proposed by Epstein and Hefeker (2003). In particular, if both players hire

a delegate (case (d)), then

q =

(1 + y1)z1

(1 + y1)z1+ (1 + y2)z2

(1)

if z1+ z2> 0, whereas q = 1/2 if z1+ z2= 0. Observe that (1) has the

following attractive properties: (i) both instruments are complementary to

each other, i.e. the second instrument yi reinforces the effect of the first

instrument zi; (ii) delegate i does not have to use the second instrument

(i.e. we can have yi= 0) - she will only use it if doing so positively affects

her expected payoff; (iii) if both delegates do not use their second instru-

ment, then (1) reduces to the standard Tullock contest success function

q = z1/(z1+z2). Notice that if we would have in (1) the term yiziinstead of

the term (1 + yi)zi, then we would always have that yi= ziin equilibrium,

which is not very interesting. See further in Epstein and Hefeker (2003, p.

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83). Proceeding, if only one player, say player i, does not hire a delegate

(cases (b) and (c)), then we replace in (1) (1 + yi)ziwith ei. In the same

spirit, if both players do not hire a delegate (case (a)), then q = e1/(e1+e2)

if e1+e2> 0, while q = 1/2 if e1+e2= 0; i.e. then the situation boils down

to the standard contest of Tullock (1980).2

We will analyse the (subgame-perfect pure-strategy Nash) equilibria. Us-

ing backward induction, we first investigate for each of the cases (a) to (d)

(i.e. for given delegation decisions), the corresponding equilibrium efforts in

stage 3 and the equilibrium contracts in stage 2 (if relevant). Next, com-

bining the results of these four cases, we derive the equilibrium delegation

decisions in stage 1.

Case (a). In this case both players compete for the prize themselves in

stage 3. Given ej> 0, player i maximizes his expected payoff, i.e. he solves

?

with i ?= j and i,j = 1,2. This case corresponds to the standard Tullock

(1980) contest. So, it is well-known that the equilibrium efforts of both

players are ea

max

ei≥0

ei

ei+ ej

?

V − ei,

(2)

1= ea

2= V/4, while the expected payoffs are πa

Case (b). Now the contestants in stage 3 are delegate 1 and player 2. In

stage 3, given 0 ≤ γ1≤ 1 and e2> 0, delegate 1 solves

?

while, given z1> 0, player 2 solves

?

The next lemma characterizes the equilibrium in stage 3.

1= πa

2= V/4.

max

y1≥0,z1≥0

(1 + y1)z1

(1 + y1)z1+ e2

?

γ1V − y1− z1,

(3)

max

e2≥0

e2

(1 + y1)z1+ e2

?

V − e2.

(4)

Lemma 1 Consider case (b). For each 0 ≤ γ1≤ 1 there is a unique equi-

librium in the corresponding contest between delegate 1 and player 2 in stage

3. The following holds in this equilibrium:

(i) Delegate 1 uses one instrument if and only if 1/γ1≥

corresponding efforts are yb

√V − 1. The

1= 0 and

zb

1=

γ2

1V

(1 + γ1)2,eb

2=

γ1V

(1 + γ1)2.

(5)

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(ii) Delegate 1 uses both instruments if and only if 1/γ1<

corresponding efforts are yb

√V − 1. The

1= zb

1− 1 and

zb

1=

√V −1

γ1,eb

2=

1

γ1

√V −1

γ2

1

.

(6)

Hence, in case (b) delegate 1 uses both instruments if and only if, given

V > 1, the size of γ1is large enough. Delegate 1 never uses both instruments

if V ≤ 1.

Turning to stage 2, we present the next result.

Lemma 2 Consider case (b). Define V0≡1

the following holds:

4(√2 − 1)−4≈ 8.49. In stage 2

(i) If 0 < V ≤ V0, then there is a unique equilibrium in stage 2, in which

γb

equilibrium in stage 3. The corresponding expected payoffs of player 1

and player 2 are:

1=

√2 − 1. Delegate 1 will use one instrument in the subsequent

πb

1= (√2 − 1)2V,πb

2=V

2.

(7)

(ii) If V > V0, then there is a unique equilibrium in stage 2, in which

γb

equilibrium in stage 3. The corresponding expected payoffs of player 1

and player 2 are:

1= V−1

4. Delegate 1 will use two instruments in the subsequent

πb

1= (V

1

4− 1)2√V ,πb

2=

√V .

(8)

Lemma 2 shows that if V is larger than the threshold V0, then for player 1 it

is profitable to offer delegate 1 a contract that induces her to compete with

both instruments in stage 3. For smaller values of V , the contract offered

to delegate 1 induces her to compete with one instrument only.

Case (c). Now the contestants in stage 3 are player 1 and delegate 2.

Obviously, this is the counterpart of case (b), and we can state results similar

to those of Lemma 1 and Lemma 2 by interchanging the indices 1 and 2. In

particular, if 0 < V ≤ V0, then there is an equilibrium in stage 2, in which

delegate 2 uses one instrument, with the following expected payoffs for the

players 1 and 2:

πc

2,

1=V

πc

2= (√2 − 1)2V.

(9)

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If V > V0, then there is an equilibrium in stage 2, in which delegate 2 uses

both instruments, with associated expected payoffs

√V ,πc

πc

1=

2= (V

1

4− 1)2√V .

(10)

Case (d). Now the contestants in stage 3 are delegate 1 and delegate 2. The

probability that player 1 wins the prize is given by (1). Given 0 ≤ γ1≤ 1

and z2> 0, the problem considered in stage 3 by delegate 1 is

?

while, given 0 ≤ γ2≤ 1 and z1> 0, the problem of delegate 2 reads

?

Investigating the situation in stage 3, we may assume without loss of gen-

erality that γ2≥ γ1> 0 and present the following result.

Lemma 3 Consider case (d). For each 0 ≤ γ1≤ 1 and 0 ≤ γ2≤ 1, with

γ2= kγ1> 0 and k ≥ 1, there is a unique equilibrium in the contest between

the two delegates in stage 3. The following holds in this equilibrium:

max

y1≥0,z1≥0

(1 + y1)z1

(1 + y1)z1+ (1 + y2)z2

?

γ1V − y1− z1,

(11)

max

y2≥0,z2≥0

(1 + y2)z2

(1 + y1)z1+ (1 + y2)z2

?

γ2V − y2− z2.

(12)

(i) Both delegates use one instrument if and only if γ1V ≤ (1 + k)2/k2.

The corresponding efforts are yd

1= 0, yd

2= 0 and

zd

1=

γ2

1γ2V

(γ1+ γ2)2,zd

2=

γ1γ2

(γ1+ γ2)2.

2V

(13)

(ii) Delegate 1 uses one instrument whereas delegate 2 uses two instru-

ments if and only if (1 + k)2/k2< γ1V ≤ (1 + k2)2/k2. The corre-

sponding efforts are yd

1= 0, yd

γ2)?γ1V − (γ1

2= zd

2− 1 and

zd

1= (γ1

γ2)2,zd

2=

?γ1V −γ1

γ2.

(14)

(iii) Both delegates use both instruments if and only if γ1V > (1+k2)2/k2.

The corresponding efforts are yd

1= zd

1− 1, yd

2= zd

2− 1 and

zd

1=

γ3

1+ γ2

1γ2

2V

(γ2

2)2,zd

2=

γ2

1+ γ2

1γ3

2V

(γ2

2)2.

(15)

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Lemma 3 shows that, for given k ≥ 1, both delegates use one instrument if

the contingent fee γ1V is small, whereas both delegates use both instruments

if γ1V is large. For intermediate values of γ1V , only delegate 2 – who has a

larger contingent fee than delegate 1 – uses both instruments.

Notice that the effort levels (13) are identical to the equilibrium effort

levels that would result in a standard Tullock contest between delegate 1 and

delegate 2, i.e. if they only would be able to compete with the instruments

z1and z2. We further remark that Epstein and Hefeker (2003) also present

the results of part (iii) of Lemma 3. However, they do not discuss those of

part (ii).3

Proceeding with stage 2, we present Lemma 4. In turns out that in a

number of situations the equilibrium in stage 2 is not unique. However, fol-

lowing common practice, we can obtain uniqueness in those cases by focusing

on Pareto dominant equilibria. We call an equilibrium Pareto dominant if

there does not exist another equilibrium in which both players are strictly

better off.

Lemma 4 Consider case (d). Define V1 ≡ 3/(√3 − 1)4≈ 10.45 and let

V2≈ 12.45 be the unique positive root of ((3V )

2 the following holds:

1

2 +9)4−27V3= 0. In stage

(i) If 0 < V ≤ V1, then there is a unique Pareto dominant equilibrium,

in which γd

Both delegates use one instrument in the subsequent equilibrium in

stage 3. The corresponding expected payoffs of player 1 and player 2

are:

πd

3,

1= γd

2=1

3. (If 0 < V ≤ 8, this is the unique equilibrium.)

1=V

πd

2=V

3.

(16)

(ii) If V1 < V ≤ V2, then there are two Pareto dominant equilibria. In

the first one, we have γd

one instrument and delegate 2 uses both instruments in the subsequent

equilibrium in stage 3. The corresponding expected payoffs of player 1

and player 2 are:

1=

1

3and γd

2= (3V )−1

4. Delegate 1 uses

πd

1= 2 × 3−5

4× V

3

4,πd

2=

?

1 − (3V )−1

4

?2V.

(17)

In the second one, we have γd

both instruments and delegate 2 uses one instrument in the subsequent

equilibrium in stage 3. The corresponding expected payoffs of player 1

1= (3V )−1

4 and γd

2=1

3. Delegate 1 uses

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and player 2 are:

πd

1=

?

1 − (3V )−1

4

?2V,πd

2= 2 × 3−5

4× V

3

4.

(18)

(iii) If V > V2, then there is a unique equilibrium, in which γd

Both delegates use both instruments in the subsequent equilibrium in

stage 3. The corresponding expected payoffs of player 1 and player 2

are:

πd

4,

1= γd

2=1

2.

1=V

πd

2=V

4.

(19)

Using the above lemmas we turn to stage 1 and present our main result.

Proposition 1 Let V1≈ 10.45 and V2≈ 12.45 be as defined in Lemma 4.

We then have the following with respect to the equilibria of the model:

(i) If 0 < V < 9, then there is a unique Pareto dominant equilibrium, in

which both players do not hire a delegate in stage 1. (If 0 < V ≤ 8,

this is the unique equilibrium.)

(ii) If 9 ≤ V ≤ V1, then there exist a unique Pareto dominant equilibrium,

in which both players hire a delegate in stage 1. In this equilibrium

both delegates use one instrument in stage 3, and their contingent fees

areV

3.

(iii) If V1 < V < 16, then there exists a unique Pareto dominant equi-

librium, in which both players do not hire a delegate in stage 1. (If

V2< V < 16, this is the unique equilibrium.)

(iv) If V = 16, then there exist equilibria in which both players, one player,

or zero players hire a delegate in stage 1. If a delegate is hired, she

uses two instruments in stage 3. If both delegates are hired, then their

contingent fees areV

fee is (√2 − 1)V .

(v) If V > 16, then there is a unique equilibrium, in which both player

1 and player 2 hire a delegate in stage 1. In this equilibrium both

delegates use both instruments in stage 3, and their contingent fees

areV

3; if only one delegate is hired, then her contingent

2.

The proposition shows that, depending on the size of the prize V , we have

equilibria in which either no, one or both players hire a delegate. It is in-

teresting to compare this with the corresponding standard benchmark case,

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where we have a contest in which delegates have only one instrument at

their disposal. We recall from our discussion of W¨ arneryd (2000) that in

that case there is always a unique equilibrium in which the players do not

hire a delegate; in fact, in that case not hiring a delegate is always a dom-

inant strategy for each player in stage 1. Hence, introducing the option

that delegates use more than one instrument completely changes the set of

possible equilibria.

We further see from Proposition 1 that the decision to hire a delegate

does not depend in a monotone way on the size of V . In particular, we

have (Pareto dominant) equilibria with delegation for both 9 ≤ V ≤ V1and

V > 16. This can be understood in an intuitive way as follows. First, for

the case with relatively ’small’ values of V , i.e. 9 ≤ V ≤ V1, we obtain a

unique Pareto dominant equilibrium in which both players hire a delegate.

We observe that in this equilibrium both delegates compete ’modestly’, in

the sense that they use only one instrument. Second, consider the case

with ’intermediate’ values of V , i.e. V1< V < 16, and suppose that both

players hire a delegate in this case. It follows from the proof of Proposition 1

(see the Appendix) that we then have two subcases: (a) if V1< V ≤ V2,

then one delegate uses one instrument whereas the other delegate competes

‘aggressively’ and uses two instruments; (b) if V2 < V < 16, then both

delegates compete ’aggressively’ and use two instruments. It turns out that

in subcase (a) it is profitable for the player associated with the modest

delegate, to deviate unilaterally and not hire a delegate. Similarly, in subcase

(b) a unilateral deviation by any player is profitable. In both subcases the

prize is not large enough to sustain an equilibrium with delegation by both

players.Third, take the case with relatively ’large’ values of the prize,

i.e. V > 16. We then have a unique equilibrium in which both players

hire a delegate and where each delegate competes aggressively by using

two instruments. Now V is so large that the aggressive behaviour of both

delegates can be sustained in equilibrium.

Finally, it also follows from the proof of Proposition 1 (see the Appendix)

that in the borderline case with V = 16, each player obtains a payoff equal

to 4 in all possible situations. This explains part (iv) of Proposition 1.

3 Conclusion

We have analysed delegation in a two-person rent-seeking contest where

delegates can use two instruments whereas the players themselves can only

compete with one instrument. We endogenized both the decision to hire a

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delegate and the contingent fee offered to the delegates. It turns out that,

depending on the size of the contested prize, we have (Pareto dominant)

equilibria in which no, one or both players hire a delegate. Interestingly, the

number of players that hire a delegate in equilibrium does not depend in a

monotone way on the size of the contested prize. It depends in a non-trivial

way on the interplay between the size of the contested prize and the decision

of the delegates to compete with either one or two instruments.

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Notes

1Baik (2006) similarly analyses a two-player contest with compulsory delegation where

the delegates and players have identical abilities. In his model the payment schemes for

the delegates consist of an endogenously determined contingent fee and a nonnegative

fixed fee. Baik shows that in equilibrium the fixed fee is set equal to zero.

2Recalling our discussion of abilities in Section 1, we mention here for completeness

the following. Take a standard two-player contest, and let the probability that player 1

wins the prize be p = λe1/(λe1+ e2), where λ > 1 is a given parameter. Then player 1

has a larger ability than player 2. Note that p > 1/2 if e1 = e2.

3We remark that Epstein and Hefeker (2003, Corollary 1) present a sufficient condition

such that one contestant uses one instrument while the other contestant uses both instru-

ments. Part (ii) of our Lemma 3 presents a necessary and sufficient condition. Epstein

and Hefeker also do not give (14).

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References

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delegation. Southern Economic Journal, forthcoming

Baik, K.H. & Kim, I.-G. (1997). Delegation in contests. European Journal

of Political Economy, 13, 281–298

Epstein, G.S. & Hefeker, C. (2003).

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Liston-Heyes, C. (2001). Setting the stakes in environmental contests. Jour-

nal of Environmental Economics and Management, 41, 1–12

Lockard, A.L. & Tullock, G. (Eds.) (2001). Efficient rent-seeking: Chronicle

of an intellectual quagmire. (Boston: Kluwer Academic Publishers)

Nitzan, S. (1994). Modelling rent-seeking contests. European Journal of

Political Economy, 10, 41–60

Schoonbeek, L. (2002). A delegated agent in a winner-take-all contest. Ap-

plied Economics Letters, 9, 21–23

Schoonbeek, L. (2004). Delegation in a group-contest. European Journal of

Political Economy, 20, 263–272

Szymanski, S. (2003). The economic design of sporting contests. Journal of

Economic Literature, 41, 1137–1187

Tullock, G. (1980). Efficient rent seeking. (in J.M. Buchanan, R.D. Tollison,

& G. Tullock (Eds.), Toward a theory of the rent-seeking society (pp. 97–

112). College Station: Texas A&M University Press.)

W¨ arneryd, K. (2000). In defense of lawyers: moral hazard as an aid to

cooperation. Games and Economic Behavior, 33, 145–158

Lobbying contests with alternative

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Appendix: Proofs

Proof of Lemma 1 – There is no equilibrium in which z1= 0 and/or e2=

0. Hence, the Kuhn-Tucker (KT) conditions characterizing an equilibrium

are

z1e2γ1V

((1 + y1)z1+ e2)2≤ 1 with an equality sign if y1> 0,

(1 + y1)e2γ1V

((1 + y1)z1+ e2)2= 1,

(1 + y1)z1V

((1 + y1)z1+ e2)2= 1.

(A.1)

(A.2)

(A.3)

First, consider yb

and zb

zb

KT-conditions then imply yb

(6). Using zb

1= 0. We then see from the KT-conditions that zb

2. Substituting the latter equality, we find (5). The condition

1≤ 1 can be rewritten as 1/γ1≥

1= zb

1of (6), we derive that yb

1≤ 1

1= γ1eb

√V − 1. Second, consider yb

1−1 and zb

1> 0. The

1= γ1eb

2. Substituting, we obtain

1> 0 if and only if 1/γ1<√V − 1. ?

Proof of Lemma 2 – We present the proof in 3 steps. Using backward

induction, we analyse in steps 1 and 2 the cases where delegate 1 uses,

respectively, both instruments and one instrument in stage 3. We combine

results in step 3.

— Step 1: If delegate 1 uses both instruments in stage 3, it follows from

part (ii) of Lemma 1 that q = 1−γ−1

2 now becomes

max

1V−1

2. The problem of player 1 in stage

0≤γ1≤1(1 − γ−1

1= V−1

4 − 1)2√V . Next, we must guarantee that the condition

mentioned in part (ii) of Lemma 1 is satisfied. We observe that 1/γb

√V −1 holds with the value of γb

i.e. if and only if V > (1+√5

2

)4≈ 6.85. Remark that now the expected payoff

of player 2 equals πb1

— Step 2: If delegate 1 uses one instrument in stage 3, part (i) of Lemma 1

implies that q = γ1/(γ1+1). In turn, the problem faced by player 1 in stage

2 is

max

γ1+ 1)(1 − γ1)V.

1V−1

2)(1 − γ1)V.

(A.4)

In the optimum γb

equals πb1

4. The corresponding expected payoff of player 1

1 = (V

1

1<

1just derived if and only if V

1

2−V

1

4−1 > 0,

2=√V .

0≤γ1≤1(

γ1

(A.5)

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It follows that in the optimum we have γb

payoff of player 1 is πb2

condition mentioned in part (i) of Lemma 1 is satisfied with this value of γb

Observe that 1/γb

Remark that now the expected payoff of player 2 is πb2

— Step 3: We see that for 0 < V ≤ (1+√5

V > 2/(√2−1)2, πb2

V ≤ 2/(√2 − 1)2, we obtain that πb1

?

1= −1 +√2, while the expected

1= (√2 − 1)2V . Next, we must guarantee that the

1≥√V − 1 holds if and only if V ≤ 2/(√2 − 1)2≈ 11.66.

1.

2= V/2.

2

)4, πb1

1is not relevant, while for

1and πb2

1if and only if V >1

1is not relevant. Comparing πb1

1> πb2

1for (1+√5

4(√2 − 1)−4.

2

)4<

Proof of Lemma 3 – There cannot exist an equilibrium in which z1= 0

and/or z2= 0. Consequently, the KT-conditions characterizing an equilib-

rium are

(1 + y2)z1z2γ1V

((1 + y1)z1+ (1 + y2)z2)2≤ 1 with an equality sign if y1> 0,(A.6)

(1 + y1)(1 + y2)z2γ1V

((1 + y1)z1+ (1 + y2)z2)2= 1,

(1 + y1)z1z2γ2V

((1 + y1)z1+ (1 + y2)z2)2≤ 1 with an equality sign if y2> 0,(A.8)

(1 + y1)(1 + y2)z1γ2V

((1 + y1)z1+ (1 + y2)z2)2= 1.

(A.7)

(A.9)

Recall that γ2= kγ1with k ≥ 1. Hence, in equilibrium we can only have

three situations: both delegates use one instrument; delegate 1 uses one

instrument whereas delegate 2 uses both instruments; or both delegates use

both instruments.

First, suppose that both delegates use only one instrument, i.e. yd

yd

Using the latter equality, we find zd

that zd

γ1V ≤ (1 + k)2/k2. The condition on γ1V given in part (i) follows from

k ≥ 1. This proves part (i).

Second, suppose that delegate 1 uses one instrument, i.e. yd

delegate 2 uses both instruments. The KT-conditions then yield zd

zd

and zd

from the fact that zd

yd

Third, suppose that both delegates use both instruments.

1=

2γ1.

2= 0. The KT-conditions then show that zd

1≤ 1, zd

2given in part (i). Next, remark

2≤ 1 if and only if

2≤ 1 and zd

1γ2= zd

1and zd

1≤ 1 if and only if γ1V ≤ (1 + k)2/k, and zd

1= 0, whereas

1≤ 1,

2= 1 + yd

2presented in part (ii). The condition on γ1V given in part (ii) follows

1≤ 1 can be rewritten as γ1V ≤ (1 + k2)2/k2, while

2> 0 is equivalent to γ1V > (1 + k)2/k2. This establishes part (ii).

2and zd

1γ2= zd

2γ1. Using the latter two equalities, we derive zd

1

The KT-

15