Abstract
This article examines the impact of the distribution of preferences on equilibrium behavior in conflicts modeled as allpay auctions with identitydependent externalities. Centrists and radicals are defined using a willingnesstopay criterion that admits preferences more general than a simple ordering on the line. Extremism, characterized by a higher per capita expenditure by radicals than centrists, may persist and generate higher aggregate expenditure by radicals, even when they are relatively small in number. Our results demonstrate the importance of the institutions of conflict in determining the role of extremism and moderation in economic, political, and social environments.
This is a preview of subscription content, access via your institution.
Notes
 1.
Linster (1993) argues that such a solution exists unless the contest is degenerate in the sense that players are indifferent to the outcome.
 2.
Konrad (2006) examines the effect of silent shareholdings in an allpay auction framework with complete information and finds that the social value may increase or decrease depending on the identity of the firm that holds a share in its competitor. However, Konrad does not further analyze settings in which three firms are active in equilibrium and allows only one player’s valuation to be endogenous.
 3.
For instance, in firstprice winnerpay auctions, Funk (1996) and Jehiel and Moldovanu (1996) show that multiple payoff nonequivalent equilibria may arise. Jehiel and Moldovanu (1996) show that if players can commit in a preauction stage not to participate, both potential winners and losers may choose nonparticipation, despite the inability to avoid the negative externality. Janssen and Moldovanu (2004) show that revenue and efficiency may be unrelated to each other.
 4.
 5.
In this case we can employ our analysis to a transformed bid , \(\beta =C(b)\). We elaborate on other potential assumptions on cost in Sect. 3.
 6.
This definition is based on Siegel (2009) but accounts for the identitydependent externalities.
 7.
To our knowledge no formal definition of a radical in a general nplayer environment exists in the literature. Intuitively, Definition 1 captures the notion that if a radical player is removed from the contest, the maximal reach across all remaining pairs of players should strictly decrease. That is, the maximal willingness to pay to win decreases.
 8.
Generally the concept of social welfare additionally takes expenditure into account. We follow Jehiel and Moldovanu (2006) and Linster (1993) by using the sum of valuations to measure social welfare in a context of contests with identitydependent exernalities. This interpretation implies that the players’ expenditures are considered transfers. In some conflicts which are covered by our model, e.g. political lobbying, expenditures are often more accurately viewed as a social waste of resources. Therefore, we additionally discuss the effects of the auction CSF on expected total expenditure. We further elaborate on this issue in the conclusion.
 9.
Even if the valuation vectors were slightly perturbed in a way such that players 1 and 3 were no longer symmetric the main conclusions of Propositions 3 and 4 continue to hold in the sense that the player who was previously the radical will always continue to actively participate in the conflict and for each player who was previously a centrist there exists an equilibrium in which he stays out of the conflict. This is true even though one of the players previously a centrist is a radical in the perturbed game (see Klose and Kovenock 2014).
 10.
 11.
In this type of contest a group’s probability of winning depends on their total effort and all members of the winning group receive their valuation of winning (e.g. Baik (1993), Baik et al. (2001), Esteban and Ray (1999)). Alternatively groups may compete for a privategood prize, whereupon the prize is allocated within the winning group through a second stage contest (e.g. Katz and Tokatlidu (1996), Konrad and Kovenock (2009)) or a previously determined sharing rule (Baik and Lee 2001).
 12.
See, however, Klose and Kovenock (2014) for an analysis of allpay auctions with identitydependent externalities and more general preference structures.
 13.
We let \(p\{\hbox {i wins }\, b_j\}\) denote the probability that player \(i\) wins conditional on the event that player \(j\) bids \(b_j\).
References
Baik KH (1993) Effort levels in contests: the publicgood prize case. Econ Lett 41(4):363–367
Baik KH, Kim IG, Na S (2001) Bidding for a groupspecific publicgood prize. J Public Econ 82(3):415–429
Baik KH, Lee S (2001) Strategic groups and rent dissipation. Econ Inq 39(4):672–684
Baye M, Kovenock D, DeVries C (1996) The allpay auction with complete information. Econ Theory 8:291–305
Che YK, Gale I (1998) Caps on political lobbying. Am Econ Rev 88(3):643–651
Esteban J, Ray D (1999) Conflict and distribution. J Econ Theory 87:379–415
Fudenberg D, Tirole J (1987) Understanding rent dissipation: on the use of game theory in industrial organization. Am Econ Rev 77:176–183
Funk P (1996) Auctions with interdependent valuations. Int J Game Theory 25:51–64
Janssen M, Moldovanu B (2004) Allocation mechanisms and postallocation interaction. In: Janssen M (ed) Auctioning public assets. Cambridge University Press, Cambridge
Jehiel P, Moldovanu B (1996) Strategic nonparticipation. Rand J Econ 27:84–98
Jehiel P, Moldovanu B (2001) Efficient design with interdependent valuations. Econometrica 69(5):1237–1259
Jehiel P, Moldovanu B (2006) Allocative and informational externalities in auctions and related mechanisms. In: Blundell Richard, Newey Whitney, Persson Torsten (eds) The Proceedings of the 9th world congress of the econometric society. Cambridge University Press, Cambridge
Katz E, Tokatlidu J (1996) Group competition for rents. European Journal of Political Economy 12:599–607
Klose B, Kovenock D (2014) The allpay auction with complete information and identitydependent externalities. Econ theory. doi:10.1007/s0019901408485
Konrad KA (2006) Silent interests and allpay auctions. Int J Ind Organ 24:701–713
Konrad KA (2009) Strategy and dynamics in contests. Oxford University Press, Oxford
Konrad KA, Kovenock D (2009) The alliance formation puzzle and capacity constraints. Econ Lett 103(2):84–86
Linster BG (1993) A generalized model of rentseeking behavior. Public Choice 77(2):421–435
Maskin ES (2000) Auctions, development, and privatization: efficient auctions with liquidityconstrained buyers. Eur Econ Rev 44(4–6):667–681
Moldovanu B, Sela A (2001) The optimal allocation of prizes in contests. Am Econ Rev 91(3):542–558
Osborne MJ, Rosenthal JS, Turner MA (2000) Meetings with costly participation. Am Econ Rev 90(4):927–943
Siegel R (2009) Allpay contests. Econometrica 77(1):71–92
Siegel R (2010) Asymmetric contests with conditional investments. Am Econ Rev 100(5):2230–2260
Skaperdas S (1998) On the formation of alliances in conflict and contests. Public Choice 96(1/2):25–42
Tullock G (1967) The welfare costs of tariffs, monopolies, and theft. Econ Inquiry 5(3):224–232
Tullock G (1980) Efficient rent seeking. In: Buchanan JM, Tollison R, Tullock G (eds) Toward a theory of the rent seeking society. Texas A&M Press, College Station
Acknowledgments
We have benefited from the helpful comments of Jack Barron, Kai A. Konrad, Wolfgang Leininger, Benny Moldovanu, and seminar participants at Erasmus University Rotterdam, the Max Planck Institute for Tax Law and Public Economics, Purdue University, TU Dortmund, the University of Bonn, the University of California at Riverside, the University of York and the University of Zurich. Earlier versions of this paper were presented at the 2010 Verein für Socialpolitik Congress in Kiel, the 2010 Association for Public Economic Theory Conference in Istanbul, the 2010 European Association for Research in Industrial Economics Annual Conference in Istanbul, the 2010 Society for the Advancement in Economic Theory Conference in Singapore, and the 2011 Conference on Tournaments, Contests, and Relative Performance Evaluation at North Carolina State University. Bettina Klose gratefully acknowledges the financial support of the European Research Council (ERC Advanced Investigator Grant, ESEI249433) and the Swiss National Science Foundation (SNSF 100014 135257). Dan Kovenock has benefited from the financial support of the Social Science Research Center Berlin (WZB) and the Max Planck Institute for Tax Law and Public Economics in Munich.
Author information
Affiliations
Corresponding author
Appendices
Appendix 1: Proof of Proposition 1
Proof
In a first step we show existence by constructing an equilibrium, we then show uniqueness of the equilibrium described before in a second step involving multiple lemmas.
The strategy profile in which \(2\) stays out completely (puts mass 1 on zero) and players \(1\) and \(3\) randomize uniformly over \([0,r_{jk}]\) (\(j,k\in \{1,3\}\), \(j\ne k\)) is a Nash equilibrium. Assume that \(2\) uses the strategy \(F_2(x)=\left\{ \!\!\begin{array}{cll}0&{}\quad \hbox { for }&{}x<0\\ 1&{}\quad \hbox { for }&{}x\ge 0\end{array}\right. \). Then (by Baye et al. 1996) it is optimal for players \(1\) and \(3\) to randomize over \([0,r_{jk}]\) according to
Given that \(1\) and \(3\) apply this strategy player \(2\)’s payoff if he submits a strictly positive bid \(x\in (0,r_{2j}]\) is:
It is therefore a best response for player 2 to stay out of the conflict.
Next we prove the uniqueness of the equilibrium described in Proposition 1, by first showing that any equilibrium of \(\varGamma _{21}\) is symmetric in the sense that both radicals (players 1 and 3) choose identical strategies. In a second step we then show that the set of symmetric equilibria of \(\varGamma _{21}\) is a singleton, given by the cutthroat competition equilibrium described above.
Let \(\underline{s}_i\) and \(\bar{s}_i\) be the lower and upper bound, respectively, of the support of an equilibrium strategy for player \(i\), \(i\in I\), and define \(\bar{s}=\max _{i\in I}\{\bar{s}_i\}\). In the following, the indices \(j\) and \(k\) refer to two different radical players, i.e. \(j\!\in \!\{1,3\}, k\!\in \!\{1,3\}\backslash \{j\}\).
\(\square \)
Lemma 1
In any equilibrium of \(\varGamma _{21}\), both radicals actively participate in the conflict.
Proof
By way of contradiction, assume that one of the radical players stays out of the conflict; without loss of generality let that player be player \(1\), i.e. \(F_1(0)=1\). Given player \(1\)’s strategy players \(2\) and \(3\) would randomize up to \(r_{23}=r_{32}<r_{31}=r_{13}\). Player \(1\)’s payoff if he bids zero will be in the interval \((v_{13},v_{12})\) and he could strictly improve upon this by bidding \(r_{23}\) which would guarantee him a payoff of \(v_{11}r_{23}=v_{11}r_{12}=v_{12}\).\(\square \)
Lemma 2
\(\underline{s}_i=0\) for all \(i\in I\), and for at least one player \(l\in I\), \(F_l(0)=0\).
Proof
Assume \(\underline{s}_i>\underline{s}_l\ge \underline{s}_m\ge 0\) for some \(i,l,m\in I\). Any bid \(x\in [0,\underline{s}_i)\) results in a loss with certainty. Therefore, players \(l\) and \(m\) do not put mass anywhere over \((0,\underline{s}_i)\). Moreover, no player \(l\) or \(m\) can place a mass point at \(\underline{s}_i\), because if two or more players had a mass point at \(\underline{s}_i\), then one could improve by moving mass up, and if only one player had a mass point at \(\underline{s}_i\), then he would improve by moving the mass down. Altogether players \(l\) and \(m\) do not put mass anywhere over \((0,\underline{s}_i]\), but then player \(i\) would improve by moving mass down. This contradiction implies that there exist mutually different \(i,l,m\in I\) such that \(\underline{s}_i=\underline{s}_l\ge \underline{s}_m\ge 0\). Assume that \(\underline{s}_i=\underline{s}_l> \underline{s}_m\ge 0\) for some \(i,l,m\in I\). It cannot be the case that both players, \(i\) and \(l\), have a mass point at \(\underline{s}_i=\underline{s}_l\) (otherwise one could improve by moving mass up slightly), but then at least one of them would win with probability arbitrarily close to zero in some neighborhood above \(\underline{s}_i\) and would be better off by moving mass down to zero. It follows that in equilibrium \(\underline{s}_1=\underline{s}_2=\underline{s}_3=\underline{s}\). It cannot be the case that all three players have a mass point at \(\underline{s}\) otherwise a player could improve by moving this mass up slightly. Therefore, at least one player loses with certainty at \(\underline{s}\). Altogether this shows that \(\underline{s}_1=\underline{s}_2=\underline{s}_3=0\).\(\square \)
Lemma 3
There are no mass points at \(x\) in any player’s equilibrium distribution \(\forall x\in (0,\bar{s}]\).
Proof
Suppose player \(i\in I\) has a mass point at \(x\in (0,\bar{s}]\). Since, from Lemma 2, \(F_l(x)>0\) for every \(l\in I\), for sufficiently small \(\epsilon >0\) no player \(j\ne i\) would place mass in \((x\epsilon ,x]\) since that player could improve his payoff by moving mass from that interval to infinitesimally above \(x\). But then it is not optimal for \(i\) to put mass at \(x\).\(\square \)
Lemma 4
\(\bar{s}_1=\bar{s}_3>\bar{s}_2\).
Proof
Obviously, it cannot be the case that \(\bar{s}_i>\bar{s}_l\ge \bar{s}_m\) for some \(i,l,m\in I\), because player \(i\) would strictly improve his payoff by moving mass from \(\left( \frac{1}{2}(\bar{s}_l+\bar{s}_i),\bar{s}_i\right] \) down to \(\frac{1}{2}(\bar{s}_l+\bar{s}_i)\). Suppose, \(\bar{s}_1=\bar{s}_2=\bar{s}_3=\bar{s}>0\). Since any bid \(b_2>r_{2j}\) of player 2 is strictly dominated by \(b_2=0\) it follows that \(\bar{s}\le r_{2j}\). By Lemma 2, \(\underline{s}_i=0\) for all \(i\in I\) and at most two players may have a mass point at zero. Therefore, there exists a radical player \(j\), who is outbid with certainty when bidding zero and whose payoff from bidding zero is \(u_j^*(0)=\alpha v_{j2}+(1\alpha )v_{jk}\) for some \(\alpha \in (0,1)\). By assumption \(r_{jk}>r_{j2}\) which implies by definition that \(v_{jk}<v_{j2}\). Then, (by Lemma 2) player \(j\)’s expected equilibrium payoff would be \(u_j^*<v_{j2}\). On the other hand, by submitting a bid \(\bar{s}+\epsilon \) greater than \(\bar{s}\) player \(j\) would receive \(u_j^*(\bar{s}+\epsilon )=v_{jj}\bar{s}\epsilon \ge v_{jj} r_{j2}\epsilon =v_{j2}\epsilon \). Therefore, by choosing \(\epsilon >0\) small enough, he would improve his payoff. Thus, \(\bar{s}_1=\bar{s}_2=\bar{s}_3\) cannot hold true. By the same argument it cannot be the case that \(\bar{s}_j<\bar{s}_2=\bar{s}_k=\bar{s}\). Hence, \(\bar{s}_j=\bar{s}_k>\bar{s}_2\).\(\square \)
Lemma 5
\(\bar{s}_2<r_{2j}, j\in \{1,3\}\).
Proof
By Lemma 4 player \(2\) loses with strictly positive probability at \(\bar{s}_2\). Suppose \(\bar{s}_2\ge r_{2j}\), then player \(2\)’s equilibrium payoff at \(\bar{s}_2\) is
This is a contradiction, because player 2 could guarantee himself a payoff of at least \(v_{2j}\) by bidding zero.\(\square \)
Lemma 6
Players \(j\in \{1,3\}\) earn expected equilibrium payoffs \(v_{jj}\bar{s}\).
Proof
From Lemmas 3 and 4 players \(1\) and \(3\) must earn their expected equilibrium payoff at the upper bound of the support of their mixed strategies, \(\bar{s}\), and neither has a mass point at \(\bar{s}\). Therefore, their expected equilibrium payoff is \(u_j^*=v_{jj}\bar{s}\).\(\square \)
Lemma 7
\(F_1(x)=F_3(x)\) for all \(x\in [\bar{s}_2,\bar{s}]\).
Proof
Notice that \(F_2(x)=1\) for all \(x\in [\bar{s}_2,\bar{s}]\), and \(F_1(\bar{s})=F_3(\bar{s})=1\). From Lemma 4, for \(x\in (\bar{s}_2,\bar{s}]\)
By Lemma 6 it follows that
and by Assumption 2 (symmetric interagent antagonism) follows that players \(j\) and \(k\) use identical strategies \(F_j(x)=F_k(x)=1\frac{\bar{s}x}{r_{jk}}\) over the interval \((\bar{s}_2,\bar{s}]\). If \(\bar{s}_2>0\), then by Lemma 3 this holds over \([\bar{s}_2,\bar{s}]\). If \(\bar{s}_2=0\) rightcontinuity of \(F_i, i\in I,\) implies \(F_1(0)=F_3(0)\).\(\square \)
Lemma 8
For any nondegenerate interval \([\underline{t},\bar{t}]\in [0,\bar{s}]\) (\(\underline{t}<\bar{t}\)) there are at least two players, \(i,j\in I\), such that \(F_l(\bar{t})F_l(\underline{t})>0\) for \(l=i,j\).
Proof
Suppose there is a \(t>\underline{t}\) such that \(F_i(t)F_i(\underline{t})=0\) for all \(i\in I\), and let \(\bar{t}\) be the supremum over all \(t\) with this property, i.e. define \(\bar{t}=\sup \{t>\underline{t} : F_i(t)F_i(\underline{t})=0 \hbox { for all } i\in l\}.\) Notice that by Lemma 2 \(\underline{t}>0\). Since \(\bar{t}>\underline{t}\ge 0\) no player has a mass point at \(\bar{t}\) by Lemma 3. Let player \(i\in I\) and \(m,l\in I\backslash \{i\}\), then player \(i\)’s payoff from a bid \(\bar{t}+\epsilon \) is
On the other hand player \(i\)’s payoff from bidding \(\underline{t}\) is
which is strictly greater than \(u_i(\bar{t}+\epsilon ,F_l,F_m)\) for \(\epsilon >0\) sufficiently small. Thus, for small enough \(\epsilon >0\) a player would improve his payoff by moving mass from \([\bar{t},\bar{t}+\epsilon ]\) to \(\underline{t}\). Therefore, no \(t>\underline{t}\) such that \(F_i(t)F_i(\underline{t})=0\) for all \(i\in I\) exists.
Suppose that there is only one player \(i\in I\) with \(F_i(\bar{t})F_i(\underline{t})>0\), and denote the other two players by \(l,m\in I\backslash \{i\}\). Note that for players \(p\in \{l,m\}\), \(f_p(t)=0\) for all \(t\in (\underline{t},\bar{t})\) and \(F_p(\underline{t})=F_p(t)=F_p(\bar{t})\) for all \(t\in (\underline{t},\bar{t})\). Player \(i\)’s expected payoff from a bid \(t\in (\underline{t},\bar{t})\) is
Therefore, player \(i\) could improve his payoff by moving mass from the interval \((\underline{t},\bar{t}]\) to its lower bound \(\underline{t}\).\(\square \)
Lemma 9
\(F_1(x)=F_3(x)\) for all \(x\in [0,\bar{s}]\).
Proof
If \(\bar{s}_2=0\) then \(F_1(x)=F_3(x)\) for all \(x\in [0,\bar{s}]\) by Lemma 7, thus we assume in the following that \(\bar{s}_2>0\).
For any bid \(b_j>0\) in the support of player \(j\)’s equilibrium strategy his expected payoff must be equal to \(v_{jj}\bar{s}\) (by Lemma 6). That is:^{Footnote 13}
Define \(\alpha ,\beta , \gamma \) such that \(\alpha \equiv r_{12}=r_{21}=r_{32}=r_{23}\), \(\beta \equiv r_{13}=r_{31}\), and \(\gamma =\alpha \beta \). Note that \(\alpha , \beta >0\) and \(\gamma <0\). Then for any \(b_j,b_k\in (0,\bar{s}]\):
and
where equality must hold in A.1.1 for bids \(b_j\) in the support of player \(j\)’s equilibrium strategy and in A.1.2 for \(b_k\) in the support of player \(k\)’s equilibrium strategy.
By way of contradiction, assume that there exists some \(b_0>0\) such that \(F_1(b_0)\ne F_3(b_0)\). By Lemma 3 \(F_1\) and \(F_3\) are continuous everywhere on \((0,\bar{s}]\) and by Lemma 7 \(F_1(\bar{s}_2)=F_3(\bar{s}_2)\). This implies that either there exists an interval \([x,y]\subset (0,\bar{s}_2]\) such that \(F_1(x)=F_3(x)\), \(F_1(y)=F_3(y)\), and \(F_1(b)\ne F_3(b) \forall b\in (x,y)\), or there exists \(\bar{x}>b_0\) such that \(F_1(b)=F_3(b)\; \forall b\ge \bar{x}\) and \(F_1(b)\ne F_3(b) \forall b\in [0,\bar{x})\).
Suppose that \([x,y]\) is an interval such that \(F_1(x)=F_3(x)\), \(F_1(y)=F_3(y)\), and \(F_1(b) \ne F_3(b) \forall b\in (x,y)\). We treat the following four cases separately:

1.
\( x,y\in supp_j\cap supp_k \), where \(supp_i\) denotes the support of player \(i\)’s equilibrium strategy. Without loss of generality let \(F_j(b)>F_k(b)\) for all \(b\in (x,y)\). In this case by (A.1.1) and (A.1.2) at \(b=y\)
$$\begin{aligned} \gamma \cdot \int _{y}^{\bar{s }}F_k(s) f_2(s) ds +\beta \cdot \left( 1F_2(y)F_k(y)\right)&= \gamma \cdot \int _{y}^{\bar{s}}F_j(s) f_2(s) ds\\&\quad +\,\beta \cdot \left( 1F_2(y)F_j(y)\right) \end{aligned}$$if and only if
$$\begin{aligned} \gamma \cdot \int _{y}^{\bar{s }}(F_j(s)F_k(s)) f_2(s) ds =\beta F_2(y)\cdot \underbrace{\left( F_j(y)F_k(y)\right) }_{=0}=0 \end{aligned}$$By definition \(\gamma <0\), hence
$$\begin{aligned}\int _{y}^{\bar{s }}(F_j(s)F_k(s)) f_2(s) ds =0. \end{aligned}$$Similarly, at \(b=x\)
$$\begin{aligned} \int _{x}^{\bar{s }}(F_j(s)F_k(s)) f_2(s) ds =0. \end{aligned}$$Then, by
$$\begin{aligned} \int _{x}^{\bar{s }}(F_j(s)\!\!F_k(s)) f_2(s) ds\!=\!\int _{x}^{y}(F_j(s)\!\!F_k(s)) f_2(s) ds\!+\!\int _{y}^{\bar{s }}(F_j(s)\!\!F_k(s)) f_2(s) ds \end{aligned}$$follows that
$$\begin{aligned} \int _{x}^{y}(F_j(s)F_k(s)) f_2(s) ds =0. \end{aligned}$$If \(f_2(s)>0\) for any \(s\in (x,y)\) this contradicts \(F_j(s)>F_k(s) \forall s\in (x,y)\). If \(f_2(s)=0\) for all \(s\in (x,y)\), then by Lemma 8 \([x,y]\subseteq supp_j\cap supp_k\) and \(F_2(x)=F_2(y)\). In this case (A.1.1) and (A.1.2) simplify to
$$\begin{aligned} \bar{s}b_j&=\gamma \cdot \int _{y}^{\bar{s }}F_k(s) f_2(s) ds +\beta \cdot \left( 1F_2(y)F_k(b_j)\right) , \hbox { and}\\ \bar{s}b_k&=\gamma \cdot \int _{y}^{\bar{s }}F_j(s) f_2(s) ds +\beta \cdot \left( 1F_2(y)F_j(b_k)\right) \end{aligned}$$respectively for all \(b_j,b_k\in (x,y)\). Notice that in both expressions the integral is constant in the player’s own bid. Since \(F_j\) and \(F_k\) coincide at \(x\) and \(y\), \(\int _{y}^{\bar{s }}F_k(s) f_2(s) ds= \int _{y}^{\bar{s }}F_j(s) f_2(s) ds \). This shows that \(F_j(b)=F_k(b)\; \forall b\in (x,y)\), which contradicts our assumption.

2.
\( y\in supp_j\cap supp_k, x\in supp_j\backslash supp_k .\) Then by (A.1.1) and (A.1.2) at \(b=x\)
$$\begin{aligned}&\gamma \cdot \int _{x}^{\bar{s }}F_k(s) f_2(s) ds +\beta \cdot \left( 1F_2(x)F_k(x)\right) \le \gamma \cdot \int _{x}^{\bar{s}}F_j(s) f_2(s) ds\\&\quad +\beta \cdot \left( 1F_2(x)F_j(x)\right) \\&\Leftrightarrow \gamma \cdot \int _{x}^{\bar{s }}(F_k(s)F_j(s)) f_2(s) ds \le \beta F_2(x)\cdot \underbrace{\left( F_k(x)F_j(x)\right) }_{=0}=0 \end{aligned}$$By definition \(\gamma <0\), hence
$$\begin{aligned}\int _{x}^{\bar{s }}(F_k(s)F_j(s)) f_2(s) ds \ge 0. \end{aligned}$$If \(f_2(s)>0\) for any \(s\in (x,y)\) this implies \(F_j(s)< F_k(s)\), because by assumption \(F_j(b)\ne F_k(b)\, \forall b\in (x,y)\) and by Lemma 3 (no mass points) \(F_j\) and \(F_k\) are continuous. By assumption \(x\not \in supp_k\). So there exists an \(\epsilon >0\) such that \(F_k(x+\delta )=F_k(x)\) for all \(\delta \) such that \(0<\delta <\epsilon .\) But then
$$\begin{aligned} F_j(x+\delta )<F_k(x+\delta )=F_k(x)=F_j(x), \end{aligned}$$which is a contradiction, because \(F_j\) is a cumulative distribution function and as such is nondecreasing. If \(f_2(s)=0\, \forall s\in (x,y)\), then from Lemma 8 \([x,y]\subseteq supp_j\cap supp_k\), a contradiction to the assumption \(x\not \in supp_k\).

3.
\( x\in supp_j\cap supp_k, y\in supp_j\backslash supp_k .\) By (A.1.1) and (A.1.2) at \(b=y\)
$$\begin{aligned}&\gamma \cdot \int _{y}^{\bar{s }}F_k(s) f_2(s) ds +\beta \cdot \left( 1F_2(y)F_k(y)\right) \le \gamma \cdot \int _{y}^{\bar{s}}F_j(s) f_2(s) ds\\&\quad +\beta \cdot \left( 1F_2(y)F_j(y)\right) \\&\Leftrightarrow \gamma \cdot \int _{y}^{\bar{s }}(F_k(s)F_j(s)) f_2(s) ds \le \beta F_2(y)\cdot \underbrace{\left( F_k(y)F_j(y)\right) }_{=0}=0 \end{aligned}$$By definition \(\gamma <0\), hence
$$\begin{aligned}\int _{y}^{\bar{s }}(F_k(s)F_j(s)) f_2(s) ds \ge 0. \end{aligned}$$By (A.1.1) and (A.1.2) at \(b=x\)
$$\begin{aligned} \begin{array}{c} \displaystyle \gamma \cdot \int _{x}^{\bar{s }}F_k(s) f_2(s) ds +\beta \cdot \left( 1F_2(x)F_k(x)\right) =\gamma \cdot \int _{x}^{\bar{s}}F_j(s) f_2(s) ds\\ \displaystyle +\beta \cdot \left( 1F_2(x)F_j(x)\right) \\ \displaystyle \Leftrightarrow \gamma \cdot \int _{x}^{\bar{s }}(F_j(s)F_k(s)) f_2(s) ds =\beta F_2(x)\cdot \underbrace{\left( F_j(x)F_k(x)\right) }_{=0}=0.\\ \end{array} \end{aligned}$$\(\gamma <0\), hence
$$\begin{aligned}&\int _{x}^{y}(F_j(s)F_k(s)) f_2(s) ds+\underbrace{\int _{y}^{\bar{s }}(F_j(s)F_k(s)) f_2(s) ds }_{\le 0} =0\\&\quad \Rightarrow \int _{x}^{y}(F_j(s)F_k(s)) f_2(s) ds \ge 0 \end{aligned}$$If \(f_2(s)>0\) for any \(s\in (x,y)\), then this implies \(F_j(s)\ge F_k(s)\). By assumption, \(F_j(b)\ne F_k(b)\, \forall b\in (x,y)\), thus \(F_j(b)> F_k(b)\, \forall b\in (x,y)\). By assumption \(y\not \in supp_k\). Hence, there exists an \(\epsilon >0\) such that \(F_k(y\delta )=F_k(y)\, \forall 0<\delta <\epsilon .\) But then
$$\begin{aligned}F_j(y\delta )>F_k(y\delta )=F_k(y)=F_j(y),\end{aligned}$$a contradiction to the fact that \(F_j\) is a cumulative distribution function and as such is nondecreasing. If \(f_2(s)=0\, \forall s\in (x,y)\), then from Lemma 8 \([x,y]\subseteq supp_j\cap supp_k\), a contradiction to the assumption \(y\not \in supp_k\).

4.
\( x\in supp_j\backslash supp_k, y\in supp_k\backslash supp_j .\) By (A.1.1) and (A.1.2) at \(b=x\)
$$\begin{aligned}&\gamma \cdot \int _{x}^{\bar{s }}F_k(s) f_2(s) ds +\beta \cdot \left( 1F_2(x)F_k(x)\right) \le \gamma \cdot \int _{x}^{\bar{s}}F_j(s) f_2(s) ds\\&\quad +\beta \cdot \left( 1F_2(x)F_j(x)\right) \\&\Leftrightarrow \gamma \cdot \int _{x}^{\bar{s }}(F_k(s)F_j(s)) f_2(s) ds \le \beta F_2(x)\cdot \underbrace{\left( F_k(x)F_j(x)\right) }_{=0}=0 \end{aligned}$$By definition \(\gamma <0\), hence
$$\begin{aligned}\int _{x}^{\bar{s }}(F_k(s)F_j(s)) f_2(s) ds \ge 0. \end{aligned}$$A similar argument shows that at \(b=y\)
$$\begin{aligned}\int _{y}^{\bar{s }}(F_k(s)F_j(s)) f_2(s) ds \le 0. \end{aligned}$$Consequently,
$$\begin{aligned} \begin{aligned} 0\le&\int _{x}^{\bar{s }}(F_k(s)F_j(s)) f_2(s) ds \\=&\int _{x}^{y}(F_k(s)F_j(s)) f_2(s) ds +\underbrace{\int _{y}^{\bar{s }}(F_k(s)F_j(s)) f_2(s) ds}_{\le 0}\\ \Rightarrow&\int _{x}^{y}(F_k(s)F_j(s)) f_2(s) ds \ge 0. \end{aligned} \end{aligned}$$If \(f_2(s)>0\) for any \(s\in (x,y)\), then this implies \(F_k(s)\ge F_j(s)\). By assumption \(F_j(b)\ne F_k(b)\, \forall b\in (x,y)\), thus \(F_k(b)> F_j(b)\, \forall b\in (x,y)\). By assumption \(x\not \in supp_k\). Hence, there exists an \(\epsilon >0\) such that \(F_k(x+\delta )=F_k(x)\, \forall 0<\delta <\epsilon .\) But then
$$\begin{aligned}F_j(x+\delta )<F_k(x+\delta )=F_k(x)=F_j(x),\end{aligned}$$a contradiction to the fact that \(F_j\) is a cumulative distribution function and as such is nondecreasing. If \(f_2(s)=0 \,\forall s\in (x,y)\), then from Lemma 8 \([x,y]\subseteq supp_j\cap supp_k\), a contradiction to the assumption \(x\not \in supp_k, y\not \in supp_j\).
Taking these four possible cases together, there cannot exist any interval \([x,y]\) with \(F_1(x)=F_3(x)\), \(F_1(y)=F_3(y)\), and \(F_1(b)\ne F_3(b)\, \forall b\in (x,y)\).
Assume now that there exists an \(\bar{x}>b_0\) such that \(F_j(b)=F_k(b)\, \forall b\ge \bar{x}\) and \(F_j(b)> F_k(b), \forall b\in [0,\bar{x})\). Players \(1\) and \(3\) must earn their equilibrium payoff at (or arbitrarily close to) zero, so by (A.1.1) and (A.1.2)
If \(f_2(s)=0\) for all \(s\in (0,\bar{x})\), then (A.1.3) simplifies to
This implies \(F_2(0)=0\), which is a contradiction, because Lemma 2 and \(f_2(s)=0\) for all \(s\in (0,\bar{x})\) imply \(F_2(0)>0\).
If \(f_2(s)>0\) for some \(s\in (0,\bar{x})\), then \(\beta >0\) and \(\gamma <0\) imply that
a contradiction to (A.1.3). Consequently, there can exist no \(b_0>0\) such that \(F_1(b_0)\ne F_3(b_0)\).\(\square \)
Lemma 10
\(F\equiv F_1=F_3\) first order stochastically dominates \(F_2\).
Proof
If \(\bar{s}_2=0\), then \(F_2(x)=1\,\forall x\ge 0\). Hence, \(F\) first order stochastically dominates \(F_2\).
Therefore, assume in the following that \(\bar{s}_2>0\). By way of contradiction assume that there exists some \(b_0\in [0,\bar{s}_2)\) such that \(F_2(b_0)<F(b_0)\). Note that by Lemmas 8 and 9 \(supp_j=[0,\bar{s}], j\in \{1,3\}\). Furthermore, by Lemma 4 \(F_2(\bar{s}_2)>F(\bar{s}_2)\) and by Lemma 3 no player’s equilibrium strategy has a mass point at any strictly positive bid. Then, there must exist an interval \([\underline{t},\bar{t}\,]\subseteq (0,\bar{s}_2]\) such that \([\underline{t},\bar{t}\,]\subseteq \bigcap _{i\in I} supp_i\), \(F_2(\underline{t})<F(\underline{t})\), and \(F_2(\bar{t})>F(\bar{t}).\) \([\underline{t},\bar{t}\,]\subseteq supp_2.\) Therefore, player 2 must earn his equilibrium expected payoff at any bid \(x\in [\underline{t},\bar{t}\,];\) that is, for every \(x\in [\underline{t},\bar{t}\;]\)
where the last equality follows from Lemma 2. Hence,
Similarly, \([\underline{t},\bar{t}\,]\subseteq supp_j, j\in \{1,3\},\) implies that player \(j\), \(j\in \{1,3\}\), must earn his equilibrium expected payoff at any bid \(x\in [\underline{t},\bar{t}\,]\). Player \(j\)’s expected payoff from a bid, \(x\in [\underline{t},\bar{t}\,]\), is
Player \(j\)’s payoff must be constant on \([\underline{t},\bar{t}]\), that is,
This yields the following linear first order differential equation, which must hold for all \(x\in [\underline{t},\bar{t}\,]\)
Since \(F\) takes the form described in (A.1.4), the solution to (A.1.5) is
where \(c\in \mathbb {R}\) is a constant of integration.
By assumption \(\beta >\alpha \), thus there exists a \(\delta >0\) such that \(\beta =(1+\delta )\alpha \) and we can write
By differentiating (A.1.6) we obtain
Suppose \(F(\bar{t})<F_2(\bar{t})\), then by continuity of the equilibrium strategies (Lemma 3) \(F(\bar{t}\epsilon )<F_2(\bar{t}\epsilon )\) for sufficiently small \(\epsilon >0\). Considering \(x=\bar{t}\epsilon \) in (A.1.6) yields the necessary condition \(c>0.\) Using this in (A.1.7) shows that \(F_2'(x)<F'(x)\) for \(x\in [\underline{t},\bar{t}]\). Hence, \(F(\underline{t})<F_2(\underline{t}),\) a contradiction to the assumption that \(F(\underline{t})>F_2(\underline{t}).\) Therefore, there exists no point \(b_0\in [0,\bar{s}_2]\) such that \(F_2(b_0)<F(b_0)\), and \(F\) first order stochastically dominates \(F_2\). \(\square \)
Lemma 11
\(F_2(x)=1 \quad {\text{ for } \text{ all }} \; x\ge 0\)
Proof
Lemmas 2, 9, and 10 together imply \(F(0)=0\), hence by Lemma 2 player 2’s expected payoff in equilibrium is \(v_{2j}\). By way of contradiction assume that \(\bar{s}_2>0\). Then, by the same argument as in the proof of Lemma 10 equation (A.1.6) must hold at every \(x\in supp_2\) with \(F(x)=\left( \frac{x}{\alpha }\right) ^\frac{1}{2}\). Player 2 may not randomize over strictly positive bids arbitrarily close to zero. Indeed, if such randomization did occur, because all players’ equilibrium strategies are continuous over \((0,\bar{s}]\) by Lemma 3, \(F(0)=0\) and therefore
and \(F_2(0)<1\) (under the assumption that \(\bar{s}_2>0\)), then (A.1.6) would imply that \(c=0\), which is a contradiction to Lemma 10. Given that player 2 does not randomize over strictly positive bids arbitrarily close to zero, there exists a \(\underline{t}>0\) such that \(\underline{t}=\inf \{t>0\, t\in supp_2\}.\) Then, \(F_2(\underline{t})=F_2(0)\). By (A.1.1) player \(j\)’s expected payoff from a bid \(x\in (0,\underline{t}]\) is
\(x\in (0,\underline{t}]\) is a best response for player \(j\) therefore \(u_j^*(x,F_2,F)\) must be constant over \((0,\underline{t}]\). It follows that \(F'(x)=\frac{1}{F_2(0)\beta }\) for \(x\in (0,\underline{t}]\). From \(F(0)=0\) follows that players \(1\) and \(3\) randomize uniformly over \([0,\underline{t}]\) according to
Continuity of \(F\) at \(\underline{t}\) yields
Consequently,
Using this and \(F_2(\underline{t})=F_2(0)\) in (A.1.6) yields
which implies
This contradicts Lemma 10; therefore \(\bar{s}_2=0\).\(\square \)
Altogether, this shows that player 2 stays out of the conflict in equilibrium. Hence, the equilibrium described in Proposition 1 is the unique equilibrium of \(\varGamma _{21}\).
Appendix 2: Proof of Proposition 5
Proof
Under the assumption that all three players make positive bids with strictly positive probability and players \(1\) and \(3\) use identical strategies, i.e. \(F_1=F_3=:F\), we know that \(\underline{s}_1=\underline{s}_2=\underline{s}_3=0\) and \(\bar{s}_2=\bar{s}_1=\bar{s}_3=: \bar{s}\). Moreover, \(\bar{s}\in (r_{jk},r_{2j})\), \(j,k\in \{1,3\}, j\ne k,\) and player \(2\) cannot have a masspoint at zero. Assume that all players randomize continuously over \([0,\bar{s}]\). All players must earn their equilibrium payoff at \(\bar{s}\), therefore player \(2\)’s expected payoff from a bid \(b\in (0,\bar{s}]\), \(u_2(b,F)=v_{22}[F(b)]^2+v_{2i}(1[F(b)]^2)\), must be \(v_{22}\bar{s}\). This yields
Player \(j\)’s payoff must be \(v_{jj}\bar{s}\). Moreover, player \(j\) chooses his equilibrium strategy such that his expected payoff, \(u_j(b,F_2,F)=b+v_{j2}+[v_{jj}v_{j2}]F(b)F_2(b)+[v_{jk}v_{j2}]\int _b^{\bar{s}}F_2(s)F'(s)ds\), is maximized. The first order condition yields the first order differential equation
Using the boundary conditions \(F_2(0)=0\) and \(F_2(\bar{s})=1\) this yields
with \(\kappa =\frac{2r_{j2}}{r_{j2}+r_{jk}}>1\) and \(\bar{s}=r_{j2}\left[ 1\left( 1\frac{1}{\kappa }\right) ^\kappa \right] \). Note that \(\bar{s}\in (r_{jk},r_{j2})\) and \(F_2\) is strictly increasing.
In order to show that this equilibrium exhibits extremism, we need to show that \(F_2(x)\le F(x)\, \forall x\). All players’ cdfs coincide for \(x<0\) and \(x\ge \bar{s}\). The centrist players put strictly positive mass on zero, thus \(F_2(0)<F(0)\). For \(x\in (0,\bar{s})\),
Therefore, \(F_2\) first order stochastically dominates \(F\).\(\square \)
Rights and permissions
About this article
Cite this article
Klose, B., Kovenock, D. Extremism drives out moderation. Soc Choice Welf 44, 861–887 (2015). https://doi.org/10.1007/s0035501408641
Received:
Accepted:
Published:
Issue Date:
Keywords
 Conflict
 Allpay auction
 Identitydependent externalities
 Radicalism
 Extremism
 Contest success function
JEL Classification
 D72
 D74
 C72
 D44