## Abstract

We prove some interesting properties of unilaterally competitive games when there are more than two players. We show that such games possess: (1) a Nash equilibrium, (2) maximin-solvability, (3) strong solvability in the sense of Nash, and (4) weak acyclicity, all in pure strategies of finite or infinite games.

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## Notes

A simpler example of UC game that is not a potential game is given by De Wolf (1999):

The solvability in the sense of Nash is an immediate consequence of strategy eliminations. For the strong solvability, note that if \(s \in E(G)\), \((s'_i,s_{-i}) \in S\), \(u_i(s'_i,s_{-i})=u_i(s)\), and \((s'_i,s_{-i}) \notin E(G)\), then IENBR cannot remove \(s'_i\), and the game cannot be IENBR-solvable; a similar reasoning applies to the necessity of strong solvability for IESDS-solvability.

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## Acknowledgements

An earlier version was presented at the UECE Lisbon Meetings in Game Theory and Applications 2018. This work is supported by JSPS Grant-in-Aid for Scientific Research (C) (KAKENHI) 25380233, 16K0355301, and 20K01549. The author thanks a co-editor and anonymous reviewers for their valuable comments.

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## Appendix

### Appendix

### 1.1 A proof of Theorem 5

In this proof, we take a viewpoint that a UC game can be seen as a collection of two-person zero-sum games, to which we then apply Theorem 1.

### Proof

Suppose that *G* is an *n*-person UC game with \(n \ge 2\) and \(E(G) \ne \emptyset\). By Lemma 1, we have that player *i* chooses \(s_i\) to maximize \(u_i(s_i,s_{-i})\) given \(s_{-i}\) and player \(-i\) (“other than *i*”) chooses \(s_{-i}\) to minimize \(u_i(s_i,s_{-i})\) given \(s_i\), and this is true for any \(i \in N\). Since minimizing \(u_i(s_i,\cdot )\) is the same as maximizing \(-u_i(s_i,\cdot )\), we can think of player *i* as playing a two-person zero-sum game \(G_i=(\{i,-i\},(S_i,S_{-i}),(u_i,-u_i))\) with player \(-i\), for any \(i \in N\). Let \(E(G_i)\) and \(M(G_i)\) be the set of equilibria of \(G_i\) and the set of maximin profiles of \(G_i\), for each \(i \in N\).

Now, let \(s \in E(G)\). It is not difficult to see that \(s=(s_i,s_{-i}) \in E(G_i)\) for every \(i \in N\) (use Lemma 1 to see the optimality of \(s_{-i}\) against \(s_i\)). Then, by Theorem 1, \(s=(s_i,s_{-i}) \in M(G_i)\) for every \(i \in N\). By the definition of maximin strategy in *G*, \(s_i\) is also a maximin strategy of player *i* in *G*, for any \(i \in N\). Hence \(s \in M(G)\).

Let then \(s \in M(G)\). By the definition of maximin strategy in *G*, we have \(s=(s_i,s_{-i}) \in M(G_i)\) for any \(i \in N\). Then, since \(E(G) \ne \emptyset\), and hence \(E(G_i) \ne \emptyset\) for any \(i \in N\), we have \(E(G_i)=M(G_i)\) for any \(i \in N\) by Theorem 1. Hence \(s=(s_i,s_{-i}) \in E(G_i)\) for any \(i \in N\), and \(s \in E(G)\), for obvious reasons. \(\square\)

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Iimura, T. Unilaterally competitive games with more than two players.
*Int J Game Theory* **49**, 681–697 (2020). https://doi.org/10.1007/s00182-020-00724-2

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DOI: https://doi.org/10.1007/s00182-020-00724-2

### Keywords

- Unilaterally competitive games
- Existence of a pure strategy equilibrium
- Maximin
- Strongly solvable games
- Weak acyclicity