The Arctic scramble: Introducing claims in a contest model

The Arctic scramble: Introducing claims in a contest model

Erik Ansink⁎Institute for Environmental Studies (IVM), VU University Amsterdam, De Boelelaan 1087, 1081 HV Amsterdam, The Netherlands

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 March 2010Received in revised form 20 June 2011Accepted 21 June 2011Available online 30 June 2011

In this paper I integrate elements from the bankruptcy literature in a resource contest model. Ina contest model, agents fight over a contested resource and their investment in 'guns'determines how much of the resource is secured by each agent. In a bankruptcy problem,agents claim a share of a contested resource, and their claims determine the allocation to eachagent. The integrated model of this paper allows the combination of guns and claims to jointlydetermine the distribution of a contested resource. The relevance of such a model is motivatedusing the Arctic scramble as a leading example. This contest over the Arctic's oil and gasreserves involves the five coastal Arctic countries, each with its own distinct territorial claim. Ipropose four contest-bankruptcy rules that integrate the most common contest success functionwith four classical bankruptcy rules. These four rules are assessed and effects of integratingclaims and guns in one rule are analysed for a resource contest model.

© 2011 Elsevier B.V. All rights reserved.

JEL classification:D63D74F51

Keywords:Contest modelBankruptcy problemConflictClaimsArctic scramble

1. Introduction

Contest models are economic models that assess the outcome of a resource contest based on the level of ‘guns’ (or effort ingeneral) that persons, firms, or countries invest (Garfinkel and Skaperdas, 2007; Konrad, 2009). Examples include litigation orarmed conflict over a piece of land, lobbying over the outcome of a public policy, or appropriation of a jointly produced good at thecost of lower production (Corchón, 2007). In standard contest models, an implicit assumption is that all agents aspire to capturethe complete resource that is at stake. This assumption is not in line with many real-life situations. Consider the example of theArctic scramble. This recent conflict over Arctic territories and its vast oil reserves involves Russia, the United States, Canada,Norway, and Denmark (Greenland). Claims to the Arctic territories date back to the beginning of the 20th century. Recently, thestruggle for this territory was renewed due to expected polar ice loss as a result of climate change, which may enable profitableextraction of oil and gas. The countries involved typically have not made territorial claims to the complete region. Instead, theyclaim only parts of the Arctic—those parts to which they consider to have a legitimate claim, for instance because of allegedunderwater connections of continental shelves and ridges (Cressey, 2008).

This example – which serves as the leading example of this paper – features exogenous claims and illustrates the prominentrole that such claims may play in a resource contest. Other examples include the river sharing problem (Ansink and Weikard,2009), claims in fisheries resources (Bess, 2001), islands (Denoon and Brams, 1997), and territorial claims (Murphy, 1990). Twofactors that may influence the impact of claims in contests are the following. First, agents may realise that the credibility of theirclaim is important for the outcome of the conflict, as in the Arctic scramble. Second, agents may realise that defending their

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secured share of the contested resource is easier for smaller shares. An open question is how claims and guns interact in a contestand, more specifically, how the presence of claims affects the outcome of a contest model.

In this paper I assess appropriate methods to model the outcome of a contest as a function of both agents' guns and their claimsto the resource. Although the role of claims has been ignored in the literature on the economics of contests, there is a largeliterature on bankruptcy problems, inwhich claims play a central role. In a bankruptcy problem, agents claim a share of a contestedresource, and their claims determine the allocation to each agent (Moulin, 2002; Thomson, 2003). The bankruptcy literature seeksto solve this problem using an axiomatic approach. Several distribution mechanisms from this literature – called “bankruptcyrules” – have wide theoretical and empirical support (Herrero and Villar, 2001) and are therefore suitable candidates forintegration into a contest model. This integration of bankruptcy rules into a contest model brings together two so far unconnectedstrands of literature. There are, however, three exceptions. First, Corchón and Dahm (2009) assessed theoretical links betweencontest models and bankruptcy problems, interpreting the former in terms of the latter. The approach in this paper is different as Iaim to integrate the two. Second, Grossman (2001) included claims in a conflict setting, with agents dividing their resourceendowments over defending their own claim and contesting the other agent's claim. This implies that agents contest the resourcebeyond the level of their own claim, implicitly contesting (or claiming) the full resource. Again, the approach in this paper isdifferent. The Arctic scramble illustrates that agents may not aspire to receive a share of the resource larger than their claim. Third,Welch (1997) integrated a contest into a bankruptcy model in order to explain firms' capital structure decisions. The argument isthat creditors aim tominimise potential ex-post lobbying and litigation costs in case of bankruptcy. In general, banks are in a betterposition than public creditors to influence bankruptcy rules ex-post. Under this assumption, such wasteful rent-seeking isprevented by awarding seniority to banks ex-ante.

In my proposed integrated approach, it is important to realise that the bankruptcy literature, with few exceptions, treats claimsas exogenous to the problem at hand. Because of this exogeneity, a solution to a bankruptcy problem is not based on strategicconsiderations or optimisation on the part of the agents. Rather, the focus in this literature is on the construction of attractive rulesto allocate the resource. In the current paper, we allow investments in guns to affect such allocations.

In a contest model, the distribution of the resource is determined by a contest success function (CSF) (Skaperdas, 1996). In abankruptcy problem, this distribution is determined by a bankruptcy rule. I propose four contest-bankruptcy rules that integratethe most common CSF (the ratio CSF) with four classical bankruptcy rules (proportional rule, constrained equal awards,constrained equal losses, and the Talmud rule). Like the four bankruptcy rules that they are based on, each of these rules satisfiesspecific properties. My aim in this paper is not to propose one specific rule. The choice for one of the rules depends on thecharacteristics of the contest to which the rule is applied.

The results of this paper allow a better understanding of the particular role of claims in contests; improving existing knowledgeon the causes of conflicts, their expected outcome, and possible factors that may prevent conflict. Specifically, I find that theinclusion of (exogenous) claims in a two-agent contest model leads to the following differences to the benchmark model withoutclaims. First, equilibrium levels of guns are weakly lower than the benchmark, and more so for higher asymmetry in claims.Second, equilibrium payoffs are not unequivocally higher than in the benchmark, though higher claims lead to higher equilibriumpayoffs. Third, due to weakly lower investments in guns, the cost of conflict in equilibrium is also weakly lower than in thebenchmark. Although the costs of contests may still be significant, this result shows that costs may be lower when claims play arole in conflict.

I proceed as follows. In Section 2, contests and bankruptcy problems are briefly introduced. In Section 3, I apply classicalbankruptcy rules from the bankruptcy literature in order to adjust contest success functions for the agents' claims. In Section 4, Ianalyse a two-agent resource contest model and assess how standard results are affected by introducing claims. In Section 5,results are applied to the Arctic scramble and in Section 6, I provide some final remarks.

2. Contests and bankruptcy problems

Consider the set of agents N={1, 2,…, n} who are competing for a resource R. The agents have claims to the resource, and havethe option to invest in guns. This situation can be modelled both as a contest and as a bankruptcy problem. I will briefly introducenotation and solution concepts for both approaches, before introducing the combined approach in Section 3.

2.1. Contest

In a contest model, agents fight over a resource or prize. The value of the resource is assumed to be public information(cf. Ryvkin, 2010). The agents can invest in guns, at a certain cost, as an input to the contest. The distribution of guns over theagents determines howmuch of the resource is secured by each agent. Formally, a contest is a pair (R, g). Each agent i∈N invests inguns gi, g=(g1, …, gn) in order to secure a portion of R, as determined by a CSF.

Contest success function. A contest success function is a mapping G: g→p that assigns to every contest a probability vectorp=(p1, …, pn), p∈R+

n , such that (a) ∑i∈Npi=1, and (b) pi(g)≥0∀ i∈N.The allocation of resources to agent i is Gi(R, g)=piR. This allocation is according to winning probabilities so that Gi(R, g) is the

share of R secured by agent i (Garfinkel and Skaperdas, 2007). Requirement (a) of the CSF imposes efficiency. Requirement (b) saysthat agents secure a non-negative portion of R. Because it fits the example of the Arctic scramble, I employ a non-probabilistic

694 E. Ansink / European Journal of Political Economy 27 (2011) 693–707

outcome whereby each agent receives a share of R, as opposed to the probabilistic outcome that is often used in modelling rent-seeking contests (cf. Long and Vousden, 1987).1

Next to the all-pay auction introduced by Hillman and Samet (1987) and the difference form (Hirshleifer, 1989), one of themost commonly used CSFs is of the so-called ratio form (Tullock, 1980; Hirshleifer, 1989):

pi gð Þ = gmi∑j∈N gmj

� � ; ð1Þ

where 0bm≤1 is a ‘decisiveness’ parameter.2 In addition to the requirements of efficiency and non-negativity in the abovedefinition, this CSF is characterised by the following five properties: Imperfect discrimination,Monotonicity, Anonymity, Luce's choiceaxiom, andHomogeneity (Skaperdas, 1996; Clark and Riis, 1998; Corchón, 2007). If the Anonymity property is dropped, then Eq. (1)is adjusted to:

pi gð Þ = αigmi

∑j∈N αjgmj

� � ; ð2Þ

where αi, αjN0, i, j∈N are constants that capture relevant differences between the agents other than their investments in guns(Clark and Riis, 1998). In Section 3 this version of the CSF will prove useful.

2.2. Bankruptcy problem

In a bankruptcy problem, a resource has to be distributed over a set of agents who have claims such that the resource isinsufficient to meet these claims (cf. Moulin, 2002; Thomson, 2003). Formally, a bankruptcy problem is a pair (R, c). Each agenti∈N has a claim ci≥0, c=(c1,…, cn) to the resource R. The interesting case is, of course, where∑i∈NciNR. The portion of R that isallocated to agent i is determined by a bankruptcy rule.

Bankruptcy rule. A bankruptcy rule is a mapping F:(R, c)→ℝn that assigns to every bankruptcy problem (R, c) an allocationvector x=(x1, …, xn), x∈ℝ+

n , such that (a) ∑i∈Nxi=R, and (b) 0≤x≤c.The allocation of resources to agent i is Fi(R, c)=xi. Requirement (a) of the bankruptcy rule imposes efficiency. Requirement (b)

says that agents receive a non-negative allocation that is bounded by their claim. Note that, consistent with each strand ofliterature, the solution of a CSF (p) is expressed as a percentage, while the solution of a bankruptcy rule (x) is expressed in absolutelevels.

Four classical bankruptcy rules are the proportional rule, the constrained equal awards rule, the constrained equal losses rule,and the Talmud rule (Herrero and Villar, 2001). The definitions of these rules are as follows.

Proportional rule (PRO). For all (R, c), there exists λN0, such that xiPRO=ciλ.Constrained equal awards rule (CEA). For all (R, c), there exists λN0, such that xiCEA=min{ci, λ}.Constrained equal losses rule (CEL). For all (R, c), there exists λN0, such that xiCEL=max{0, ci−λ}.Talmud rule (TAL). For all (R, c), there exists λN0, such that

xTALi =min

12ci;λ

� �if R≤ 1

2∑j∈N cj

� �;

ci−min12ci;λ

� �otherwise:

8>>><>>>:

In addition to the requirements of efficiency, non-negativity, and claims-boundedness in the above definitions of bankruptcyrule, these four bankruptcy rules share four appealing properties, that relate to the characterising properties of the ratio CSF inEq. (1): Claims monotonicity, Resource monotonicity, Equal treatment of equals, and Consistency (Herrero and Villar, 2001; Thomson,2003).

3. A combined approach: contest-bankruptcy rules

The properties of the ratio CSF with respect to gi are closely related to the properties of the four classical bankruptcy rules withrespect to ci. For example, Anonymity is a stronger version of Equal treatment of equals (Thomson, 2003), the (resource)Monotonicity properties are similar, and Luce's choice axiom is strongly related to Consistency. These similarities allow, for instance,

1 Note that the two approaches are equivalent under the assumption of risk neutrality.2 Note that the CSF is unspecified for the case where ∑j∈N gj=0. It is quite standard to assume pi(g)=1/n, but such a ‘peaceful’ outcome is not interesting

anyway as a small investment by one of the agents would yield him the complete resource. No agent would leave this opportunity unexploited (Garfinkel andSkaperdas, 2007).

695E. Ansink / European Journal of Political Economy 27 (2011) 693–707

to link the ratio CSF to solutions of bargaining problems with claims as is done by Corchón and Dahm (2009). Here, I am interestedin combining claims and guns in one model, such that an agent's claim and his guns jointly determine his share of the contestedresource. I call such a problem a contest-bankruptcy problem, which is denoted by a triple (R, c, g). As for bankruptcy problems, theinteresting case is, of course, where ∑i∈N ci NR. The approach for solving such a problem is by integrating the ratio CSF into abankruptcy rule. I call such a rule a contest-bankruptcy rule (CBR).

Contest-bankruptcy rule. A contest-bankruptcy rule is a mapping H:(R, c, g)→ℝn that assigns to every contest-bankruptcyproblem (R, c, g) an allocation vector y=(y1, …, yn), y∈ℝ+

n , such that (a) ∑i∈Nyi=R, and (b) 0≤y≤c.The allocation of resources to agent i is Hi(R, c, g)=yi. Requirements (a) and (b) impose efficiency, non-negativity, and claims-

boundedness.Two alternative approaches to the integration of CSFs and bankruptcy rules in a CBR can be considered. The first of these is a

contest in which first claims are announced and then a contest decides which of the agents gets awarded his full claim. Such analternative modelling approach is analogous to a contest with endogenous rents. In such models, a key result is that agentsstrategically restrain their claims in order to avoid wasteful lobbying (e.g. Epstein and Nitzan, 2004; Epstein and Nitzan, 2007).Different from the current approach, this alternative scenario assumes endogeneity of claims, which is not the type of allocationproblem considered in this paper.

The second alternative approach is to model the contest as a two-stage game. In the first stage, agents exert effort in order toincrease their ‘initial’ (exogenous) claims. In a subsequent stage, the agent's share of the resource is determined by applying abankruptcy rule to their ‘final’ claims. Although, this is an appealing modelling approach, it is not appropriate for the leadingexample of the Arctic scramble, because the coastal Arctic countries are not exerting effort to increase their claims, but rather tosupport their claims submitted to the UN, as discussed in Section 5. For this type of resource contests, CBRs are more applicable.

There are, of course, various ways to integrate the ratio CSF into a bankruptcy rule and the question now is how to construct anappropriate CBR. The approach here is to take the four classical bankruptcy rules – PRO, CEA, CEL, and TAL – and replace thesymmetric part of these rules by an asymmetric part that reflects the agents' investments in guns, according to the ratio CSF. Thefour CBRs are constructed below.3 I will show that two of these rules – the CBRs based on PRO and TAL – are identical to functionalforms that have been used in contest models before, which illustrates the intuitive appeal of combining guns and claims in onemodel. This integration of previously used functional forms is one justification for the approach used here. A second justification isthat this approach preserves the properties of the CSF and bankruptcy rules when respectively claims or guns are equal, asdiscussed below.

3.1. PRO

The CBR based on PRO assigns each agent a share of the resource proportional to the product of agents' claims and guns, subjectto no agent receiving more than his claim. Note that, if claims are equal, differences in allocation should only reflect differences inguns, according to the ratio CSF (Eq. 1). To achieve this, it is convenient to drop the anonymity property following Clark and Riis(1998) and apply the asymmetric ratio CSF displayed in Eq. (2). Differences in claims are included by setting αi=ci. In addition, acap on the allocation is included to assure that no agent is allocated more than his claim:

yPROi = min ci; cigmi λ

� �s:t:

λsolves∑j∈N

yPROj = R:

ð3Þ

For n=2 and ∑j∈NgjN0 this CBR can be relatively easily solved for λ to find that yi = min ci;max R−cj;cigmi

cigmi + cjgmjRg

( )(.

Again, as was explained in footnote 2, the CBR is unspecified for the case where ∑j∈Ngj=0. It would be sensible to assume thatthe allocation is then given by the bankruptcy rule that the CBR is based on. Recall, however, that such a ‘peaceful’ outcome is notinteresting anyway as a small investment by one of the agents would yield him the complete resource. I will therefore omit thispossibility throughout the paper, for all CBRs. In addition, in Section 4 we will assume only strictly positive investments in guns.

This CBR based on PRO allows for agents with an equal amount of guns to secure different shares of the resource. Rosen (1986),for instance, uses the αi parameters to model differences in agents' abilities in sports or career games (without the claimsboundedness constraint). This functional form is also present in the literature on rent-seeking with different relative abilities andpossibly asymmetric valuation of the rent (cf. Stein, 2002). In the current paper, the parameters reflect agents' claims butotherwise serve a similar objective. Note that for larger n, the number of elements in the function of yi equals 2n−1+1. Thisexponential increase is the reasonwhy I focus on two-agent problems in the next section, while problemswithmore agents can besolved numerically, as is done for the Arctic scramble in Section 5.

3 When interpreting the rules, recall that pi is a proportion, while xi and yi are absolute levels.


3.2. CEA

The CBR based on CEA assigns each agent a share of the resource proportional to the ratio of guns, subject to no agent receivingmore than his claim:

yCEAi = min ci;gmi

∑j∈Ngmj

!· λ

( )

s:t:λ solves ∑

j∈NyCEAj = R:

ð4Þ

For n=2 and ∑j∈N gjN0 this CBR can be relatively easily solved for λ to find that yi = min ci;max R−cj;gmi

gmi + gmjR

( )( ).

Again, for larger n, the number of elements in the function of yi equals 1+2n−1.

3.3. CEL

The CBR based on CEL assigns each agent a share of the resource such that their losses compared to their claims are inverselyproportional to the ratio of guns, subject to no agent receiving a negative share. CEL is the dual rule of CEA and the CBR based onCEL can therefore be derived similarly to Eq. (4):

yCELi = max 0; ci−∑j∈Ng

mj

gmi

!⋅λ

( )

s:t:λ solves ∑

j∈NyCELj = R:

ð5Þ

For n=2 and ∑j∈NgjN0 this CBR can be relatively easily solved for λ to find that yi = max 0; ci−max ci−R;gmj

gmi + gmjci +ð

��cj−RÞgg.

3.4. TAL

The CBR based on TAL is a combination of those based on CEA and CEL. It halves each claim and follows CEA until each half claimis met, and then it applies CEL to the remaining half claims (Moulin, 2002). Therefore, the CBR based on TAL is a combination ofEqs. (4) and (5):

yTALi =f min12ci;

gmi∑j∈Ng

mj⋅λ

( )if ∑j∈Ngj > 0 and R≤1

2∑j∈N cj

� �

max12ci; ci−

∑j∈N gmjgmi

!⋅λ

( )if ∑j∈Ngj > 0 and R >

12∑j∈N cj

� �

s:t:

λsolves∑j∈N

yTALj = R:

ð6Þ

For n=2, TAL is similar to the contested garment method (Aumann and Maschler, 1985; Herrero and Villar, 2001) so that for

∑j∈N gjN0 this CBR can be relatively easily solved for λ to find that yi = R−cj +gmi

gmi + gmjci + cj−R�

. This two-agent CBR is

equal to the functional form used by Ansink and Weikard (2009) in a contest model on river water. It provides a very intuitivecombination of a bankruptcy problemwith a contest model, because it implies that fighting only occurs over the contested part ofthe resource. To be precise, denote the contested part of the resource by R̂ = ci + cj−R. The construction of R̂ implies that bothagents claim 100% of R̂ , which allows to apply a standard contest model to the conflict with resource R̂ .

Note that integration of claims and guns in the four CBRs (Eqs. (3)–(6)) does not eradicate their respective properties.Properties of the ratio CSF remain to hold for the case of equal claims and properties of the bankruptcy rules remain to hold forthe case of equal guns. Clearly, claims and guns act as substitutes in all four rules. This substitutability makes that an agent witha high claim needs fewer guns to secure a given allocation. Also, the requirement of claims-boundedness of the CBRs impliesthat agents do not receive a share larger than their claim, no matter their effort. This threshold relates CBRs to contests withcaps on shares.


These observations on the relation between efforts and claims do not tell us whether having high or low claims is profitable,how it affects equilibrium investment in guns, and what proportion of their claim agents can secure. The answers to thesequestions are given in the following two sections, starting in the next section with a two-agent resource contest model.

To finish this section, it is informative to describe the scope of the rules proposed here. Contest-bankruptcy rules integratestrategic CSFs with axiomatic bankruptcy rules and as such the question may arise how to interpret these integrated rules. Here, Ifollow Thomson (2001) who discusses the scope of the axiomatic programme and provides two interpretations of axiomaticsolutions. In one interpretation, the axioms reflect social values so that axiomatic solutions equal normative answers to allocationproblems. In the second interpretation, axioms reflect components of behaviour so that axiomatic solutions describe actualoutcomes of strategic allocation problems. A clear example of the latter interpretation is the Nash programme, where the strategicand axiomatic solutions coincide, while “each helps to justify and clarify the other” (Nash, 1953). Many other complementaries aredescribed by Thomson (2001). Both interpretations can be directly applied to CBRs. An additional argument for the application ofaxiomatic solutions in strategic situations is given by empirical evidence on bankruptcy rules which shows that these rules areindeed descriptive of actual behaviour in allocation problems with claims (Gächter and Riedl, 2006; Bosmans and Schokkaert,2009; Herrero et al., 2010). It is therefore natural to interpret contest-bankruptcy rules as rules that describe actual outcomes ofstrategic allocation problems. A consequence of this interpretation is that there is no need for a social planner to select a contest-bankruptcy rule or to consider criteria for selecting one of the rules.4

4. CBRs in a resource contest

In this Section I use the four CBRs in a resource contest model (cf. Garfinkel and Skaperdas, 2007; Konrad, 2009). For reasonsexplained in Section 3 – an exponential increase in the number of cases to consider as n increases – I focus on the case of twoagents. This restriction, combined with the piecewise functions (3)–(6) implies discontinuities in many cases considered below.Therefore we restrict ourselves in this section to gi≥� with � small but strictly positive. Results for problems with more than twoplayers (and without this restriction) are illustrated in Section 5, using the Arctic scramble as an example.

4.1. Benchmark: no claims

In the benchmark contestmodel without claims, investment in guns yields a share of the resource, but it also brings about costs.Equilibrium investment in guns and related payoffs are easily obtained. In the resource contest model, agent i's payoff viequals

vi gi; gj� �

= pi gi; gj� �

R−gi: ð7Þ

Using Eq. (1), the FOC of this payoff function can be derived and rearranged to obtain

gmi =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimRgm−1

i gmjq

−gmj : ð8Þ

The best response function is an implicit function of this FOC. Because of symmetry of payoff functions, it is possible to simplifyEq. (8) in order to determine the equilibrium levels of guns:

g⁎i = g⁎ =m4R: ð9Þ

These equilibrium values gi⁎ can be used to calculate equilibrium payoffs, using Eqs. (1) and (7):

v⁎i = v⁎ =2−m4

� �R: ð10Þ

Eqs. (9) and (10) provide sufficient information to describe what happens to conflict intensity and payoffs when conflictparameters change. Two key results are that investments in guns are positively related to both R and m, while payoffs are

positively related to R but negatively to m. The cost of conflict equals R−2 ×2−m4

� �R = m

2 R, which increases in m and R, asexpected (Garfinkel and Skaperdas, 2007).

4 An exception is the case where agents invest in rent-seeking efforts to influence a politician. The contest could then be modelled as agents reporting theirclaims to the politician and subsequent lobbying to increase their share of the rent.


4.2. Contests with claims

When claims are introduced according to the four CBRs, claims enter the payoff function as follows:

vi gi; gj; ci; cj� �

= yi−gi: ð11Þ

Equilibrium values of guns and payoffs for the four CBRs are derived in the same way as was done for the benchmark modelwithout claims. For each CBR (Eqs. (3)–(6)), the FOC of this payoff function can be derived and rearranged to obtain bestresponses. Because three of the four CBRs are piecewise functions, their solutions are rather involved. The number of sub-domainsin these piecewise functions is limited at three – due to the two-agent setting – but would explode in case of more agents. Thesecases represent constrained solutions where one of the agents gets allocated his full claim (PRO and CEA) or nothing (CEL). Thismay remove the incentive to invest in guns. In what follows, I add the relevant rule in superscript only when confusionmay occur,for ease of notation.

4.2.1. PROUsing Eqs. (3) and (11), the payoff function is vi = min ci;max R−cj;

cigmicigmi + cjgmj

R

( )( )−gi. There are two cases:

1. Whencigmi

cigmi + cjgmjR≤R−cj then max R−cj;

cigmicigmi + cjgmj

R� �

= R−cj. The payoff function simplifies to vi=min{ci, R−cj}−

gi=R−cj−gi. We have∂v∂gi

= −1 so that gi⁎=BRi(gj)=�. This case's condition impliesci�m + cjgmj

cjgmj

!cj≤R.

2. Whencigmi

cigmi + cjgmj

!R > R−cj then max R−cj;

cigmicigmi + cjgmj

!R

( )=

cigmicigmi + cjgmj

!R. There are two sub-cases.

2a. Ifcigmi

cigmi + cjgmj

!R > ci then the payoff function simplifies to vi=ci−gi. We have ∂v

∂gi= −1 so that gi⁎=BRi(gj)=�. This

case's conditions imposeci�m + cjgmj

gmj

!N R >

ci�m + cjgmj�m

!.

2b. Ifcigmi

cigmi + cjgmj

!R≤ci then the payoff function simplifies to vi =

cigmicigmi + cjgmj

!R−gi. We have

∂v∂gi

= 0 implies that

gi⁎=BRi(gj) is the solution to gi of (cigim+cjgjm)2=cicjmRgi

m−1gjm.

Four out of six combinations of cases 1 and 2 are impossible (see the Appendix), which leaves two possible combinations:

Agent i (case 1) and agent j (case 2b). This combination of cases implies gi*= � and gj* solves (cigim+cjgjm)2=cicjmRgj

m−1gim.

Case 1's condition for agent i implies cj≤cjgmj

ci�m + cjgmj

!R. Case 2b's condition for agent j implies cj≥

cjgmjci�m + cjgmj

!R. Combining

these two implications, we obtain cj =cjgmj

ci�m + cjgmj

!R. As this equality is not necessarily satisfied by the best response of agent

j, gj*=BR(gi), this combination of cases is only possible in mixed strategies.

Agent i (case 2b) and agent j (case 2b). This combination of cases implies gi* solves (cigim+cjgjm)2=cicjmRgi

m−1gjm and gj*

solves (cigim+cjgjm)2=cicjmRgj

m−1gim. By rearranging and substitution we find symmetric investments in guns which yield

the equilibrium level of guns:

g�i = g�j =cicjmR

ci + cj� �2 : ð12Þ

This case's conditions imply ci+cj≥R. Because cicjmR�

= ci + cj� 2≤m

4R – see Eq. (9) – equilibrium levels of guns are weakly

lower than the benchmark. In addition, more asymmetric claims lead to lower gun levels.Using Eqs. (3), (11), and (12), equilibrium payoffs are

v⁎PRO

i = v⁎PRO =ci + cj−cjm� �

ciR

ci + cj� �2 : ð13Þ


For a given resource R, equilibrium payoffs are weakly higher than the benchmark – see Eq. (10) – if

ci + cj−cjm� �

ciR

ci + cj� �2 ≥ 2−m

4

� �R⇔2 c2i −c2j

� �≥−m c2i + c2j

� �+ 2mcicj: ð14Þ

This condition does not necessarily hold, indicating that the presence of claimsmay lead to lower or higher payoffs compared tothe benchmark. Note that the condition holds for agent i whenever ci≥cj; the agent with the higher claim has a higher payoff forany value ofm. Finally, given that the equilibrium levels of guns in Eq. (12) is lower than in the benchmark, it follows directly thatthe cost of conflict is also lower.

4.2.2. CEAUsing Eqs. (4) and (11), the payoff function is vi = min ci;max R−cj;

gmigmi + gmj

Rg( )

−gi

(There are two cases:

1. Whengmi

gmi + gmjR≤R−cj then max R−cj;

gmigmi + gmj

R

( )= R−cj. The payoff function simplifies to vi=min{ci, R−cj}−gi=

R −cj−gi. We have ∂v∂gi = −1 so that gi*=BRi(gj)=�. This case's condition implies

�m + gmjgmj

!cj≤R . If gj=� then cj≤ 1

2R.

2. Whengmi

gmi + gmj

!R > R−cj then max R−cj;

gmigmi + gmj

!R

( )=

gmigmi + gmj

!R. There are two sub-cases.

2a. Ifgmi

gmi + gmj

!R > ci then the payoff function simplifies to vi=ci−gi. We have

∂v∂gi

= −1 so that gi*=BRi(gj)=�. This

case's conditions impose�m + gmj

gmj

!cj > R >

�m + gmj�m

!ci. If gj=� then cj > 1

2R > ci.

2b. Ifgmi

gmi + gmj

!R≤ci then the payoff function simplifies to vi =

gmigmi + gmj

!R−gi. We have ∂v

∂gi= 0 implies that gi*=BRi(gj)

is the solution to gi of (gim+gjm)2=mRgi

m−1gjm.

Four out of six combinations of cases 1 and 2 are impossible (see the Appendix), which leaves two possible combinations:

Agent i (case 1) and agent j (case 2b). This combination of cases implies gi*=� and gj* solves (gim+gjm)2=mRgj

m−1gim. Case 1's

condition for agent i implies cj≤gmj

�m + gmj

!R. Case 2b's condition for agent 2 implies cj≥

gmj�m + gmj

!R. Combining these

two implications, we obtain cj =gmj

�m + gmj

!R. As this equality is not necessarily satisfied by the best response of agent j, gj*=

BR(gi), this combination of cases is only possible in mixed strategies.

Agent i=1 (case 2b) and agent j=2 (case 2b). This combination of cases implies gi* solves (gim+gjm)2=mRgi

m−1gjm and gj*

solves (gim+gjm)2=mRgj

m−1gim. By substitution we find the unique and symmetric equilibrium g�i = g�j =

mR4

. This case's

conditions imply ci > 12R and cj > 1

2R. This combination is equal to the benchmark situation (and independent of claims).Investment in guns, equilibrium payoffs and the cost of conflict are equal to their benchmark values.

4.2.3. CELUsing Eqs. (5) and (11), the payoff function is vi = max 0; ci−max ci−R;

gmjgmi + gmj

ci + cj−R� ( )( )

−gi. There are two cases:

1. Whengmj

gmi + gmj

!ci + cj−R�

bci−R then max ci−R;gmj

gmi + gmj

!ci + cj−R� ( )

= ci−R. The payoff function simplifies to

vi=max{0, ci− (ci−R)}−gi=− gi. We have∂vi∂gi

= −1 so that gi*=BRi(gj)= �. This case's condition impliesgmj

�m + gmj

!ci + cj−R�

bci−R. If gj=� then ciNR+cj.

2. Whengmj

gmi + gmj

!ci + cj−R�

≥ci−R then max ci−R;gmj

gmi + gmj

!ci + cj−R� ( )

=gmj

gmi + gmj

!ci + cj−R�

. There are two

sub-cases.

2a. If 0 N ci−gmj

gmi + gmj

!ci + cj−R�

then the payoff function simplifies to vi=−gi.Wehave∂v∂gi

= −1 so that gi*=BRi(gj)=�.


2b. If 0≤ci−gmj

gmi + gmj

!ci + cj−R�

then the payoff function simplifies to vi = ci−gmj

gmi + gmj

!ci + cj−R�

−gi. We have

∂v∂gi

= 0 implies that gi*=BRi(gj) is the solution to gi of (gim+gjm)2=m(ci+cj−R)gim−1gj

m.

Five out of six combinations of cases 1 and 2 are impossible (see the Appendix), which leaves one possible combination:

Agent i (case 2b) and agent j (case 2b). This combination of cases implies gi* solves (gim+gjm)2=m(ci+cj−R)gim−1gj

m and gj*solves (gim+gj

m)2=m(ci+cj−R)gjm−1gim. Note that these equalities are independent of individual claims; only the sum of claims

matters. Therefore, investments in guns are equal for both agents and by substitution we find

g�i = g�j =m4

ci + cj−R� �

: ð15Þ

This case's conditions imply ci≥cj−R and cj≥ci−R.For a given resource R, equilibrium levels of guns are weakly lower than the benchmark – see Eq. (9) – if

m4

ci + cj−R�

≤m4R.

This condition does not necessarily hold, but it does hold for the relevant case where ci, cj≤R, i.e. agents do not claimmore than theavailable resource. This indicates that for the relevant range of ci and cj, the presence of claims decreases investments in guns.

Using Eqs. (6), (11), and (15), equilibrium payoffs are

v⁎i = v⁎ = R−cj +2−m4

� �ci + cj−R� �

: ð16Þ

For a given resource R, equilibrium payoffs are weakly higher than the benchmark – see Eq. (10) – if R−cj +2−m4

� �ci + cj−R�

≥ 2−m4

� �R. This condition does not necessarily hold, indicating that agents can be worse off or better off

than in the benchmark. Finally, given that investment in guns are weakly lower than the benchmark for the relevant range of ci andcj, the cost of conflict is also weakly lower than the benchmark.

4.2.4. TALUsing Eqs. (6) and (11), the payoff function is vi = R−cj +

gmigmi + gmj

!ci + cj−R�

−gi. Note that this function can be

rearranged to vi = ci−gmj

gmi + gmj

!ci + cj−R�

−gi, which equals the only possible combination of cases for CEL. Hence, the

analysis of TAL is equal to CEL. Investment in guns, equilibrium payoffs and the cost of conflict are therefore equal too.

4.3. Symmetric equilibrium

In the symmetric case where ci = cj = c > 12R, investments in guns are symmetric too. This simplification rules out (mixed

strategy) constrained solutions and best responses can be simplified in order to determine the equilibrium levels of guns:

g⁎PRO

i = g⁎CEA

i =m4

� �R ≥ g⁎

CELi = g⁎

TALi =

m4

� �2c−Rð Þ: ð17Þ

For the case of symmetric claims, investment in guns is equal to the benchmarkmodel for both PRO and CEA, see Eq. (9). For therelevant case where ci, cj≤R we have (2c−R)≤R so that investment in guns is lower than in the benchmark model for both CELand TAL. Equilibrium payoffs are:

v⁎PRO

i = v⁎CEA

i =2−m4

� �R ≤ v⁎

CELi = v⁎

TALi =

2 + m4

� �R−1

2mc: ð18Þ

For the case of symmetric claims, payoffs are equal to the benchmark model for both PRO and CEA, see Eq. (10). Again, for the

relevant case where ci, cj≤R we have2−m4

� �R≤ 2 + m

4

� �R−1

2mc so that payoffs are higher than in the benchmark model for

both CEL and TAL. For PRO and CEA, the cost of conflict remains unchanged atm2R, increasing in m and R but independent of c.

For CEL and TAL, the cost of conflict equals R−2 ×2−m4

� �R− 1

2mc� �

= m c−12R

� �, which is positive for the relevant range

of c >12R. Note, however, that for CEL and TAL the cost of conflict increases in m and c but decreases in R. The reason for this

difference is that CEL and TAL assign losses compared to claims, so that the conflict is limited to the contested share of the resource.This contested share shrinks when R increases or when c decreases.

The implication of this result is that contest models may overestimate investments in guns when claims are ignored. Despitethe simple setting of symmetric agents, this result already demonstrates the subtle differences that may arise in selecting amongvarious ways to integrate claims and guns in one model.


4.4. Main result

So far in this section, I have assessed equilibrium levels of guns, payoffs, and the cost of conflict in two-agent contests withclaims, using the CBRs derived in Section 3. The analysis has revealed a number of similarities as well as differences between thedifferent equilibria. The main results are summarised in the following proposition.

Proposition. The inclusion of claims in a two-agent contest model using CBRs leads to the following differences with the benchmarkmodel without claims.

(a) Equilibrium levels of guns are weakly lower than in the benchmark, and more so for less symmetric claims.(b) Equilibrium payoffs are not necessarily higher than in the benchmark, depending on the relative levels of claims.(c) Due to weakly lower investments in guns, the cost of conflict in equilibrium is weakly lower than in the benchmark.

Clearly, the recognition of claims in the modelling of a resource contest reveals that such contests may be less inefficient thanconventionally thought. The level of rent dissipation (cf. Hillman, 2011) is lower than in the benchmark case, andmore so for moreasymmetric agents. This result is quite similar to results found in studies of contests with endogenous rents (e.g. Epstein andNitzan, 2004; Epstein and Nitzan, 2007) and contests with asymmetric valuation of the rent (e.g. Nti, 2004). This last similarity isnot surprising since a higher claim can be interpreted as a higher valuation for the resource as a whole.

There are no clear-cut criteria for assessing which of the four CBRs is more realistic to model a given contest. The difference inconflict costs between CBRs indicates that it may be of interest to competing agents to coordinate on the importance of claims inconflicts. Such coordination can be considered a norm or convention. Some of these conventions are more efficient than others, asis apparent from the analysis in this section. In this sense, the current analysis shows similarities with the analysis of Anbarci et al.(2002), who compared the inefficiencies arising from different norms on how to settle conflict, in terms of the applied bargainingsolution. The importance of such norms in practise has been demonstrated for instance by Dixon (1994) and Mohlin (2010).

5. The Arctic scramble

In this section I extend the two-agent case from the previous section to assess a conflict with more than two agents. Asmentioned before, this creates an exponential increase in the number of cases to consider, so that I will approach this problemnumerically. I apply the results obtained in Section 4 to the Arctic scramble. This conflict over Arctic territories, including its vast oiland gas reserves, involves five major players. These are the coastal Arctic countries Russia, the United States, Canada, Norway, andDenmark (Greenland).5 Based on expectations of polar ice loss (Serreze et al., 2007; Stroeve et al., 2011), these countries arecompeting in a struggle to obtain the rights to these resources. Ice loss will enable profitable extraction of resources that include upto a quarter of the world's undiscovered oil and gas (AMAP, 2007; Gautier et al., 2009), which provides an indication of the highstakes involved. Recently, ice loss expectations were reconfirmed by Wang and Overland (2009) who, using IPCC modelprojections, estimate that by the year 2037, the Arctic sea will face its first ice-free summer.

Countries are submitting their claims to the United Nations, as required by the 1982 UN Convention on the Law of the Sea(UNCLOS). UNCLOS is ratified by all of the above-mentioned countries except for the United States; it allows countries to makeclaims to international waters beyond their exclusive economic zone which extends 200 nautical miles into sea. Claims have to besupported by scientific evidence to support the validity of the claim based on criteria set by UNCLOS. One of these criteria, is thatthe claimed area is a ‘natural prolongation of its continental shelf’, which turns out to be very ambiguous (Cressey, 2008). A crucialongoing dispute is whether the Lomonosov ridge, that bisects the Arctic Ocean, is actually attached to one or more of Europe, Asia,or North America. As a result of this ambiguity, many of the claims are overlapping, and countries are deploying underwaterexpeditions in order to prove the validity of their claims. Nevertheless, UNCLOS is rather powerless when it comes to settlement ofthe submitted claims, as all countries, except Norway, have stated to opt out of binding dispute resolution provided by UNCLOS(Holmes, 2008).

The Arctic scramble has been interpreted both as a game that may escalate in armed conflict and as a process that willultimately end in a cooperative governance structure for the Arctic region (Young, 2009). The model developed in this paper mayshed some light on the likeliness of investments in ‘guns’ in the struggle for the Arctic's resources. I assess this conflict as a resourcecontest where five countries compete over the rights to the area beyond the countries' economic exclusive zones, as well asunsettled border conflicts within these zones. As the exact value of the Arctic's oil and gas reserves is uncertain, I normalise itsvalue to R=1, and because most of these reserves are undiscovered and speculative (AMAP, 2007), the value of the countries'claims is simplified by assuming that expected oil and gas field are evenly spread over the contested territory. Claims are derivedfrom the already submitted claims to UNCLOS of Russia and Norway6 and complemented with data on Canada, Denmark, and theUnited States from the International Boundary Research Unit (IBRU) of Durham University.7 Claims amount to cR=0.53 (Russia),cC=0.29 (Canada), cU=0.13 (US), cN=0.10 (Norway), and cD=0.07 (Denmark). These claims add up to∑ci=1.12which can beinterpreted as only 12% of the region being contested. The reason for this arguably low estimate is that IBRU uses conservativeassumptions to derive the expected claims of Canada, Denmark, and the United States. Specifically, IBRU assumes that where

5 For a recent overview of the conflict, see Byers (2009).6 See http://www.un.org/depts/los/.7 See http://www.dur.ac.uk/ibru/resources/arctic/.


http://www.un.org/depts/los/

http://www.dur.ac.uk/ibru/resources/arctic/

maritime boundaries do not yet exist, countries do not claim area beyond the so-called ‘median lines’. Given the stakes involvedand the effort that countries are investing in preparing their claims to UNCLOS, this assumption is likely to lead tounderestimations of the true claims. Nevertheless, no alternative data is currently available, as Canada, the United States, andDenmark are expected to submit their claims only in a number of years. Below, I will therefore investigate the consequences ofthese three countries submitting 20% higher bids than those assumed by IBRU. This will also add additional insight in howequilibrium gun investments and payoffs depend on the level of claims.

The four CBRs introduced in Section 3 and analysed in Section 4 for two-agent resource contests, can now be applied to theclaims data for the Arctic scramble. Results for the equilibrium levels of guns are shown in Figs. 1 and 2 for CBRs based on PRO, CEL,and TAL, for the complete range of values for ‘decisiveness’ parameter m. CEA is not included in these figures because it providesonly mixed strategy equilibria, as discussed in Section 4. In interpreting the figures for PRO, CEL, and TAL, it is instructive tocompare the results to the baseline equilibrium. In this equilibrium, gun investments increase linearly inm, from gi*(m=0)=0 togi*(m=1)=0.16 for all i. This implies that payoffs decrease from vi*(m=0)=0.2 to vi*(m=1)=0.04 for all i. Finally, the cost ofconflict increases from ∑gi*(m=0)=0 to ∑gi*(m=1)=0.8. These baseline results are depicted as dashed lines in Figs. 1 and 2.

For PRO, equilibrium gun investments are weakly higher for higher claims for all levels of m. Surprisingly, gun investments donot always increase inm. Beyond a certain level ofm and for small-claim countries, it is not worthwhile to try competing with thehigh-claim countries. Kinks in the curves occur where a country becomes constrained by its claim, so that higher investments inguns would not yield an increased share of the resource (recall that CBRs satisfy claims-boundedness). Compared to the baseline

Fig. 1. Arctic scramble: equilibrium investment in guns for Russia (R), Canada (C), United States (U), Norway (N), and Denmark (D) as a function of 'decisiveness'parameter m. Dashed lines present the baseline equilibrium without claims.


equilibrium, Fig. 1 shows that only Russia and Canada have higher gun investments for low levels of m. Payoffs are higher thanbaseline for these two countries, but lower for the US, Norway and Denmark. The cost of conflict increases from∑gi*(m=0)=0 to∑gi*(m=1)=0.33, substantially lower than baseline.

When cC, cU, and cD are increased by 20% to cC=0.348, cU=0.156, and cD=0.084 – panel (b) in Figs. 1 and 2 – there are somenotable differences. First, Norway joins Denmark by avoiding the conflict for high levels of m. Second, the ‘arms race’ betweenRussia and Canada (and the US to a lesser extent) becomes more intense. Note that the PRO solution is independent of aproportional increase in all claims (not shown in figures). This may be considered undesirable as it implies that, given such aproportional increase, there is no difference in gun investments between a situation with small and one with large claims.

For both CEL and TAL, equilibrium gun investments are substantially lower than for PRO. TAL shows a similar picture as CELbecause R = 1 > 1

2∑cj = 0:56 so that the second sub-domain of the piecewise function (7) applies, which works like CEL appliedto half-claims. Again, higher claims lead to weakly higher gun investments for all levels ofm. Surprisingly, countries are divided intwo groups depending on the level ofm. Countrieswith large claims invest equallywhile countrieswith small claimsmaynot investat all. Instead, they invest until a certain level ofm beyondwhich investing becomes too costly (Denmark in CEL and DenmarkwithNorway in TAL). Because CEL is constrained at 0 while TAL is constrained at 1

2 ci – compare Eqs. (5) and (6) – the TAL constraint is

Fig. 2. Arctic scramble: equilibrium payoffs to Russia (R), Canada (C), United States (U), Norway (N), and Denmark (D) as a function of 'decisiveness' parameterm.Dashed lines present the baseline equilibrium without claims.


more likely to be binding, so that countries stop investing earlier. Clearly, equilibrium gun investments aswell as the cost of conflictare much lower than in baseline. Payoffs are higher though, except for countries with small claims for low levels of m.

When cC, cU, and cD are increasedby20%–panels (d) and (f) in Figs. 1 and2– investments inguns increase and countrieswith smallclaims drop out sooner. Consequentially, the cost of conflict increases. Canada's payoffs increase due to its higher claim. For theUnitedStates and Denmark this increase in payoffs does not occur, because it is offset by the loss of incurring higher investment costs.

Summarising, despite the uncertainty on the exact claimsput forward by thefive countries, the CBRs applied to the Arctic scrambleshow that this conflict may be considerably less severe than when analysed with a conventional contest model. Countries with largerclaims will invest more effort in securing their claims than do small claim-countries. Consequentially, they also secure the largestshares of the Arctic territory and receive the largest payoffs. Because of their low investments in guns, countries' payoffs are not verysensitive to the level of the decisiveness parameter m. Again, there is a similarity with the literature on rent-seeking contests withasymmetric valuationof the rent. Oneof the results in these studies is thatnot all agentswill necessarily enter the contest (HillmanandRiley, 1989; Stein, 2002). This feature is nicely reflected in the Arctic scramble for all three CBRs, most clearly for CEL and TAL.

What do these results imply for the Arctic scramble? One observation is that the level of claims is a key determinant for payoffs.Countries thus have an incentive to gather as much evidence as required to submit as high claims as possible to UNCLOS. This isexactly what is occurring in reality, with all coastal Arctic countries deploying expensive underwater expeditions and geologicalsurveys to back up their claims. A second observation is that much of the literature that envisages the prospect of armed conflictover the Arctic's resources (cf. Borgerson, 2008) is misplaced. At least misplaced to the extent that the ‘conflict’will not be terriblysevere. Though much may be at stake, if one or more of the presented CBRs present a realistic story, the cost of conflict is likely tobe low due to the presence of claims.

6. Final remarks

In this paper I proposed four rules that integrate the most common contest success function with four classical bankruptcyrules. This integration allows a combination of guns and claims to jointly determine the outcome of a resource contest.Results showed that, generally, the inclusion of claims in a contest model substantially lowers the cost of conflict. This highlightsthe importance of recognising claims in analysing resource contests. The usefulness of the approach is demonstrated by anapplication to the Arctic scramble. The model, however, is relevant to a wide range of other contest settings. There are tworequirements for real-world examples to fit the model of this paper. The first one is that some agents have exogenous claims to aresource that is insufficient tomeet these claims. The second requirement is that there is a possibility for lobbying or other forms ofeffort to increase one's share. Jointly, these two requirements are not very restrictive, so that many real-world examples areapplicable.

Note that in the analysis of the Arctic scramble I have not explicitly considered spatial aspects. That is, countries do not claim apercentage of theArctic, but rathermake claims to specific areas. For instance, Canada and theUShave overlapping claims to anarea of20,000 km2 in the Beaufort Sea while Canada and Denmark disagree on their maritime boundary in the Lincoln Sea, north ofGreenland. When these spatial aspects are considered important, the Arctic scramble could be analysed as a set of independentcontests for only those areas that feature overlapping claims. Nevertheless, it is hard to see the different contested areas in isolationfrom each other. The disputes are inevitably linked in one single sovereignty dispute over the Arctic (Byers, 2009), so that the currentanalysis applies.

In other situations, there may be good reasons to divide a resource contest with claims into a set of conventional contests. Thechoice between the two is likely to be driven by factors exogenous to the model presented in this paper. Depending on theexpected costs, agents involved in such a resource contest may have an incentive to try to push the conflict into either a singlecontest with claims, or a set of smaller conventional contests.

Acknowledgements

Part of the research underlying this paper was carried out while the author was affiliated to Wageningen University. I thankHarold Houba, the editor and three anonymous reviewers for useful suggestions.

Appendix. Derivations for Section 4.2

PRO

Using Eqs. (3) and (11), we find that four out of six combinations of cases 1 and 2 are impossible:

i (case 1) and j (case 1). This combination of cases implies gi⁎=gj⁎=�. Allocations are yi=R−cj and yj=R−ci, so that yi+yj=2R−ci−cjbR, which violates the efficiency requirement. This combination is impossible.i (case 1) and j (case 2a). This combination of cases implies gi⁎=gj⁎=�. Because gj⁎=�, case 1's condition implies ci+cj≤R, acontradiction. This combination is impossible.i (case 2a) and j (case 2a). This combination of cases implies gi⁎=gj⁎= �. Subsequently, this case's condition iscj�m + ci�m

�m

� �N R >

cj�m + ci�m

�m

� �, which simplifies to ci+cjNRNci+cj, a contradiction. This combination is impossible.


i (case 2a) and j (case 2b). The condition of case 2a for agent i iscigmi

cigmi + cjgmj

!R > ci which can be rewritten as

R−ci >cjgmj

cigmi + cjgmj

!R. This condition contradicts the condition of case 2 for agent j:

cjgmjcigmi + cjgmj

!R > R−ci. This

combination is impossible.

CEA

Using Eqs. (4) and (11), we find that four out of six combinations of cases 1 and 2 are impossible:

i (case 1) and j (case 1). This combination of cases implies gi⁎=gj⁎=�. Allocations are yi=R−cj and yj=R−ci, so that yi+yj=2R−ci−cjbR, which violates the efficiency requirement. This combination is impossible.i (case 1) and j (case 2a). This combination of cases implies gi⁎=gj⁎= �. Because gj⁎=�, case 1's condition implies cj≤ 1

2R, butbecause also gi⁎=�, case 2's condition implies cj > 1

2R which contradicts. This combination is impossible.i (case 2a) and j (case 2a). This combination of cases implies gi⁎=gj⁎=�. Subsequently, this case's condition gives bothcj > 1

2R > ci and ci > 12R > cj, a contradiction. This combination is impossible.

i (case 2a) and j (case 2b). The condition of case 2a for agent i isgmi

gmi + gmj

!R > ci which can be rewritten as

R−ci >gmj

gmi + gmj

!R. This condition contradicts the condition of case 2 for agent j:

gmjgmi + gmj

!R > R−ci. This combination is

impossible.

CEL

Using Eqs. (5) and (11), we find that five out of six combinations of cases 1 and 2 are impossible:i (case 1) and j (case 1). This combination of cases implies gi⁎=gj⁎=�. Allocations are yi=0 and yj=0, so that yi+yj=0bR,

which violates the efficiency requirement. This combination is impossible.i (case 1) and j (case 2a). This combination of cases implies gi⁎=gj⁎=�. Case 1's condition for agent i implies ciNcj+R. This

condition contradicts the condition of case 2 for agent j: cibcj−R. This combination is impossible.i (case 2a) and j (case 2a). This combination of cases implies gi⁎=gj⁎= �. Subsequently, this case's condition implies both

cj− ciNR and ci− cjNR, a contradiction. This combination is impossible.

i (case 2a) and j (case 2b). The condition of case 2a for agent i is 0 N ci−gmj

gmi + gmj

!ci + cj−R�

which can be rewritten as

gmigmi + gmj

!ci + cj−R�

b cj−R. This condition contradicts the condition of case 2 for agent j:gmi

gmi + gmj

!ci + cj−R�

≥ cj−R. This

combination is impossible.i (case 1) and j (case 2b). This combination of cases implies gi⁎=� and gj⁎ solves gmj =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim ci + cj−R�

gm−1j �m

q−�m. Case 1's

condition for agent i impliesgmj

�m + gmj

!ci + cj−R�

b ci−R. This condition contradicts the condition of case 2b for agent j:gmj

�m + gmj

!ci + cj−R�

≥ ci−R. This combination is impossible.

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Documents

The Arctic scramble: Introducing claims in a contest model