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Group formation in a dominance-seeking contest

verfasst von: Dongryul Lee, Pilwon Kim

Erschienen in: Social Choice and Welfare | Ausgabe 1/2022

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Abstract

We study a group formation game. Players with different strengths form groups before expending effort to win a prize. The prize has the nature of the reward for outdoing in competition such as holding a dominant position among players or being recognized as a dominant status. So, it has the nature of public goods within a winning group (group-specific public goods). In open membership game, we find that a single player stays alone and the others form a group together in equilibrium. The stand-alone player can be anyone except for the first and second strongest players in the contest. However, strong (Nash) equilibrium predicts that the weakest player is isolated. Similarly, we find that in exclusive membership game, every structure can emerge in equilibrium but the weakest player is isolated in the strong equilibrium.

Vorheriger Artikel An analytical and experimental comparison of maximal lottery schemes

Nächster Artikel Proxy selection in transitive proxy voting

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Beginning with the seminal works of Tullock (1967, 1980), Krueger (1974), and Becker (1983), research on the contest theory has been increasing rapidly, and the literature on contests is now enormous. The following are some studies on the contest literature: Nitzan (1994), Szymanski (2003), Garfinkel and Stergios (2007), Corchón (2007), Konrad (2009), Connelly et al. (2014), Chowdhury and Gürtler (2015), Congleton and Hillman (2015), Dechenaux et al. (2015), Mealem and Nitzan (2016), Vojnović (2016), Sheremeta (2018), Corchón and Marco (2018), and Fu and Wu (2019).

A group’s performance is defined as the sum of the group members’ performances and its contest success function (CSF) is imperfectly discriminating in these studies while the followings adopt different ways of aggregating group members’ performances or/and deterministic CSF: Lee (2012), Kolmar and Rommeswinkel (2013), Chowdhury et al. (2013), Barbieri et al. (2014), Chowdhury and Topolyan (2016a, 2016b), Chowdhury et al. (2016), Lee (2017), and Lee and Song (2019).

For example, suppose that $I=\left\{ 1, 2, 3, 4\right\}$, $g_1 =1$, $g_2 =2$, $g_3 =2$, and $g_4 = 3$. Then, in the open membership game, three groups are formed and a group structure $G=\left\{ G_1 , G_2, G_3\right\}$ is derived where $G_1 =\left\{ 1\right\}$, $G_2 =\left\{ 2, 3\right\}$, and $G_3 =\left\{ 4\right\}$.

By Assumption 2, when $K=1$, every player gets zero payoff and thus does not exert any effort. In the next section, we show that the grand coalition cannot be an equilibrium outcome.

Following Epstein and Mealem (2009), our model can be extended to the case of low complementarity among members’ performances and a convex cost function. We leave it for our future works and appreciate the anonymous referee on this precious comment.

In Remark 1, we formally show that this case cannot be an equilibrium outcome. See the proof.

We can also have (10) by adding share functions for the K groups in Cornes and Hartley (2005), where the share function for $G_k$ is $s_k (X) \equiv 1-\frac{X}{B_{1k}}$ and $\sum _{k=1}^K s_k (X)=1$ with $X=\sum _{l=1}^K X_l$. Then, from equilibrium value of X ($X^*$), $X_k^*$ is given as $s_k (X^*) X^*$, $e_{1k}^*$ as $\frac{X_k^* }{a_{1k}}$, $p_k^*$ as $s_k (X^*)$, and $\pi _{1k}^*$ as $v_{1k} s_k (x^*)^2$. The conditions in (7) are equivalent to the ones for $s_k (X^*)>0$ for all $k=1, \ldots , K$. We thank the anonymous referee for suggesting the share function approach to our analysis.

Note that $B_{11} = \max \left\{ B_{11}, B_{12}, \ldots , B_{1K}\right\} =a_1 v_1$.

See the proof of Lemma 5 in Appendix A.

We can show this as well by the share function approach of Cornes and Hartley (2005). Since the share function for $G_1$ ($s_1 (\cdot )$) monotonically increases as the number of groups participating in the contest decreases, player 1k in $G_k$ can be better off by giving up his own group and joining to $G_1$ as long as his movement does not result in the grand coalition. Thus, $K=2$ should be at equilibrium. We formally present this in Appendix B, where our main results (Proposition 1, 2, and 5) are derived from the share function approach. We deeply thank the referee for sharing the outstanding proofs for Lemmas A, B, and C in the appendix.

The detailed proof is presented in Appendix A.

We appreciate the associate editor’s and the referee’s insightful comments on Proposition 1 in connection with the herding behavior of our players. Especially, the suggestion for extending our model to a sequential move game will be an important topic for our future work.

The strong equilibrium is the Coalition-Proof Nash one (Bernheim et al. 1987).

For example, suppose that $I=\left\{ 1, 2, 3, 4\right\}$, $g^1 =\left\{ 1\right\}$, $g^2 =\left\{ 1, 2, 3\right\}$, $g^3 =\left\{ 1, 2, 3\right\}$, and $g^4 =\left\{ 1, 2, 3, 4\right\}$. In game $\Gamma$, the strict unanimity is required to form a group and thus the group structure $G^{(\gamma )}=\left\{ G_1 , G_2 , G_3, G_4\right\}$ is derived where $G_1 =\left\{ 1\right\}$, $G_2 =\left\{ 2\right\}$, $G_3 =\left\{ 3\right\}$, and $G_4 =\left\{ 4\right\}$. On the contrary, in game $\Delta$, the group structure $G^{(\delta )}=\left\{ G_1 , G_2, G_3\right\}$ is derived where $G_1 =\left\{ 1\right\}$, $G_2 =\left\{ 2, 3\right\}$, and $G_3 =\left\{ 4\right\}$.

Even without the conditions in (7), we can say that at least the strongest two players in the contest, player 1 and 2, have deviation incentives from the grand coalition from Stein (2002).

For instance, suppose that $G=\left\{ G_1 , G_2, G_3 \right\}$, where $G_1 =\left\{ 1, 2, 7, 8, \ldots , n \right\}$, $G_2 =\left\{ 3, 4\right\}$, and $G_3 =\left\{ 5, 6\right\}$. This implies that, in the first stage, all the players $i \in I \backslash \left\{ 3, 4, 5, 6\right\}$ choose $g^i =\left\{ 1, 2, 7, 8, \ldots , n\right\}$, player 3 and 4 choose $g^3 =g^4 =\left\{ 3, 4\right\}$, and player 5 and 6 choose $g^5 =g^6 =\left\{ 5, 6\right\}$. From Lemma 2, player 6 will be better off if he can join $G_1$. What happens if he changes his decision from $g^6 =\left\{ 5, 6\right\}$ to $g_d^6 =\left\{ 1, 2, 6, 7, 8, \ldots , n\right\}$ for the purpose of joining $G_1$? Player 6’s deviation results in the new group structure $G^{'}=\left\{ G_1 , G_2 , G_3, G_4 \right\}$, where $G_1 =\left\{ 1, 2, 7, 8, \ldots , n \right\}$, $G_2 =\left\{ 3, 4\right\}$, $G_3 =\left\{ 5\right\}$, and $G_4 =\left\{ 6\right\}$. That is, player 6’s deviation just brings the break of his former group, and it makes him worse off than under G.

For instance, suppose that, in the first stage of the game, player 1 chooses $g^1 =\left\{ 1, 2\right\}$ and all the players $i\in I \backslash \left\{ 1\right\}$ choose $g^i =\left\{ 2, 3, \ldots , n\right\}$. Then, a group structure $G^{(\gamma )} =\left\{ G_1 , G_2 \right\}$ is determined, where $G_1 =\left\{ 1\right\}$ and $G_2 =\left\{ 2, 3, \ldots , n \right\}$. Given the group structure G, if player 2 changes his decision $g^2$ to $g_d^2 =\left\{ 1, 2\right\}$, then a new group structure $G^{(\gamma )^{'}}=\left\{ G_1 , G_2 , \cdots , G_{n-1}\right\}$ is chosen, where $G_1 =\left\{ 1, 2\right\}$ and $G_k =\left\{ k+1 \right\}$ for $k=2, 3, \ldots , n-1$.

We are grateful for the anonymous referee’s justification for assuming $g^{i}=\left\{ i\right\}$ whenever $|g_k |=1$ in another way.

We thank the anonymous referee for clarifying this point.

The detailed analysis can be provided upon request.

We again thank the associate editor for suggesting this sequential game and providing intuitive guess for its solutions.

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Titel: Group formation in a dominance-seeking contest
verfasst von: Dongryul Lee
Pilwon Kim
Publikationsdatum: 22.06.2021
Verlag: Springer Berlin Heidelberg
Erschienen in: Social Choice and Welfare / Ausgabe 1/2022
Print ISSN: 0176-1714
Elektronische ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-021-01346-7

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