Skip to main content
Log in

Games with the total bandwagon property meet the Quint–Shubik conjecture

  • Original Paper
  • Published:
International Journal of Game Theory Aims and scope Submit manuscript

Abstract

This paper revisits the total bandwagon property (TBP) introduced by Kandori and Rob (Games Econ Behav 22:30–60, 1998). With this property, we characterize the class of two-player symmetric \(n\times n\) games, showing that a game has TBP if and only if the game has \(2^{n}-1\) symmetric Nash equilibria. We extend this result to bimatrix games by generalizing TBP. This sheds light on the (wrong) conjecture of Quint and Shubik (Int J Game Theory 26:353–359, 1997) that any nondegenerate \(n\times n\) bimatrix game has at most \(2^{n}-1\) Nash equilibria. We also provide an equilibrium selection criterion to two subclasses of games with TBP.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Notes

  1. The bandwagon effect can be interpreted as a network externality and is relevant for conformity, herd behavior, and information cascade.

  2. In the late 1970s and the 1980s, consumers struggled to choose between videotape formats of VHS by JVC and Betamax sold by Sony, but as the number of the VHS users reached a threshold, the domino effect emerged and eventually VHS became predominant in the market. Other notable examples are technology adoptions for the high definition optical disc formats between Blu-ray Disc by Sony and HD-DVD by Toshiba (e.g. Fackler 2008), and for browsers in the late 1990s between Internet Explorer by Microsoft and Netscape by Navigator and in recent years between Google Chrome, Mozilla Firefox, Internet Explorer, Safari by Apple, and Opera (The Economist 2013).

  3. For instance, see Granovetter and Soong (1986), Becker (1991), Bikhchandani et al. (1992), Karni and Levin (1994), and Pesendorfer (1995) as related theoretical works; Biddle (1991) and McAllister and Studlar (1991) as empirical analysis; Plott and Smith (1999) as a relevant experiment. See Rohlfs (2001) for a comprehensive analysis on the bandwagon effect in high-tech industries.

  4. Keiding (1997) and McLennan and Park (1999) prove the conjecture in the case of \(n \le 4\) and Quint and Shubik (2002) for games where payoff matrices are identical between two players, while von Stengel (1999) shows that it no longer holds in general by constructing a general lower bound on the maximal number of Nash equilibria for even numbers of n. As its application, he provides a counterexample of a \(6 \times 6\) bimatrix game with 75 Nash equilibria, which is larger than \(2^{6}-1=63\). In the case of \(n=5\), however, we still do not know if the Quint-Shubik conjecture holds true.

  5. Using these software tools, we can consider extensive form games as well as strategic ones.

  6. Similarly, with Theorem 2 we can also consider bimatrix games, despite having an additional burden of checking if each of those \(2^n -1\) (possibly asymmetric) equilibria has a pair of different supports and both players support pair is identical (see Definition 3 and Theorem 2 in detail). This additional task is relatively easy to implement thanks to the software tools.

  7. Those methods include the “evolutionary learning method” based on the best response dynamics with mutation (Kandori et al. 1993; Young 1993); the “global game method” (Carlsson and van Damme 1993); the “incomplete information game method” (Kajii and Morris 1997); the “perfect foresight dynamics method” (Matsui and Matsuyama 1995; Hofbauer and Sorger 1999); the “spatial dominance method” (Hofbauer 1999).

  8. See also Milgrom and Shannon (1994), Topkis (1998), and Vives (1990, 2001).

  9. A pure coordination game \({\mathbf{g}}=(A,g)\) is slightly perturbed if we take an arbitrarily small \(\varepsilon _{ij} \in \mathbb {R}\) for all \(i,j \in A\), which can be positive or negative, and add it to the payoff \(g_{ij}\) by \(g_{ij} + \varepsilon _{ij}\). All slightly perturbed pure coordination games are coordination games.

  10. With our restriction to symmetric (coordination) games, we can simply define the curb set by a subset of strategies instead of a subset of strategy profiles: a subset of strategies \(S\subseteq A\) is a curb set if for any \(x\in \mathring{\varDelta }(S)\), \( \text {br}(x) \subseteq S =\cup _{x \in \mathring{\varDelta }(S)}\text {supp}(x)\). The curb set is a solution concept, whereas TBP is a condition imposed on games. Formally, a curb set is a product set of pure strategies defined in an N-person normal-form game. By definition, TBP is equivalent to the condition that any subset \(S\subseteq A\) is a curb set.

  11. This implies that all these NE are regular (Harsanyi 1973).

  12. A game \({\mathbf{g}}\) is nondegenerate if for any \(i =1,2\) and \(x \in \varDelta \), \(|\text {br}_{i}(x)|\le |\text {supp}(x)|\), and a (pure) coordination game is defined in the same way as in symmetric games.

  13. Since we focus on games with TBP, we restrict attention here to symmetric (two-player) coordination games. But the concept of the 1/2–dominant equilibrium is defined for all games.

  14. A game is generic if there is an open and dense set of the game.

  15. In fact, Monderer and Shapley (1996) define a potential function in an (possibly asymmetric) N-person game where the symmetry of potential functions does not necessarily hold. For instance, a two-player bimatrix game \({\mathbf{g}} = (A,(g^{i})_{i=1,2})\) as defined in previous section has a potential function \(v: A^2 \rightarrow \mathbb {R}\) of \({\mathbf{g}}\) if for any \(h=1,2\) and any \(i, i^{\prime }, j \in A\), \(g_{ij}^h - g_{i^{\prime }j}^h = v_{ij} - v_{i^{\prime }j}\). If a game is symmetric, by definition, we obtain the symmetry of \(v_{ij}=v_{ji}\). Hofbauer and Sigmund (1988) call such a game a (rescaled) partnership game.

  16. The literature has shown that the potential game method provides a consistent equilibrium selection with other equilibrium selection methods, such as the global game method (Frankel et al. 2003) and the incomplete information method (Ui 2001; Morris and Ui 2005; Oyama and Tercieux 2009). For the relation between those methods, see Basteck and Daniëls (2011), Honda (2011), and Oyama and Takahashi (2011).

  17. This is constructed by using a restricted version of the \(6 \times 6\) bimatrix game given by Savani and von Stengel (2006, p.418) that has 75 NE, which is more than \(2^6-1=63\).

  18. For any given \(4\times 4\) game, \(2^4-1=15\) is the maximal number of NE (Keiding 1997; McLennan and Park 1999).

  19. See, for instance, Topkis (1998) for the detail.

References

  • Basteck C, Daniëls TR (2011) Every symmetric \( 3 \times 3\) global game of strategic complementarities has noise-independent selection. J Math Econ 47:749–754

    Article  Google Scholar 

  • Basu K, Weibull JW (1991) Strategy subsets closed under rational behavior. Econ Lett 36:141–146

    Article  Google Scholar 

  • Becker GS (1991) A note on restaurant pricing and other examples of social influence on price. J Polit Econ 82:83–99

    Google Scholar 

  • Biddle J (1991) A bandwagon effect in personalized license plates? Econ Inq 29:375–388

    Article  Google Scholar 

  • Bikhchandani S, Hirshleifer D, Welch I (1992) A theory of fads, fashion, custom, and cultural change as informational cascades. J Polit Econ 100:992–1026

    Article  Google Scholar 

  • Carlsson H, van Damme E (1993) Global games and equilibrium selection. Econometrica 61:989–1018

    Article  Google Scholar 

  • Fackler M (2008) Toshiba Concedes Defeat in the DVD Battle, The New York Times

  • Frankel DM, Morris S, Pauzner A (2003) Equilibrium Sslection in global games with strategic complementarities. J Econ Theory 108:1–44

    Article  Google Scholar 

  • Granovetter M, Soong R (1986) Threshold models of interpersonal effects in consumer demand. J Econ Behav Organ 7:83–99

    Article  Google Scholar 

  • Harsanyi JC (1973) Oddness of the number of equilibrium points: a new proof. Int J Game Theory 2:235–250

    Article  Google Scholar 

  • Harsanyi JC, Selten R (1988) A general theory of equilibrium selection in games. MIT, Cambridge

    Google Scholar 

  • Hofbauer J (1999) The spatially dominant equilibrium of a game. Ann Oper Res 89:233–251

    Article  Google Scholar 

  • Hofbauer J, Sigmund K (1988) The theory of evolution and dynamical systems. Cambridge University Press, Cambridge

    Google Scholar 

  • Hofbauer J, Sorger G (1999) Perfect foresight and equilibrium selection in symmetric potential games. J Econ Theory 85:1–23

    Article  Google Scholar 

  • Honda J (2011) Noise-independent selection in global games and monotone potential maximizer: a symmetric \(3 \times 3\) example. J Math Econ 47:663–669

    Article  Google Scholar 

  • Kajii A, Morris S (1997) The robustness of equilibria to incomplete information. Econometrica 65:1283–1309

    Article  Google Scholar 

  • Kandori M, Mailath GJ, Rob R (1993) Learning, mutation, and long run equilibria in games. Econometrica 61:29–56

    Article  Google Scholar 

  • Kandori M, Rob R (1998) Bandwagon effects and long run technology choice. Games Econ Behav 22:30–60

    Article  Google Scholar 

  • Karni E, Levin D (1994) Social attributes and strategic equilibrium: a restaurant pricing game. J Polit Econ 102:822–840

    Article  Google Scholar 

  • Keiding H (1997) On the maximal number of nash equilibria in an \(n \times n\) Bimatrix game. Games Econ Behav 21:148–160

    Article  Google Scholar 

  • Leibenstein H (1950) Bandwagon, snob, and veblen effects in the theory of consumers’ demand. Quart J Econ 64:183–207

    Article  Google Scholar 

  • Matsui A, Matsuyama K (1995) An approach to equilibrium selection. J Econ Theory 65:415–434

    Article  Google Scholar 

  • McAllister I, Studlar DT (1991) Bandwagon, underdog, or projection? Opinion polls and electoral choice in Britain, 1979–1987. J Polit 53:720–741

    Article  Google Scholar 

  • McKelvey RD, McLennan AM , Turocy TL (2016) Gambit: software tools for game theory, Version 16.0.1. http://www.gambit-project.org. Accessed 10 Aug 2017

  • McLennan A, Park I-U (1999) Generic \(4 \times 4\) two person games have at most 15 nash equilibria. Games Econ Behav 26:111–130

    Article  Google Scholar 

  • Milgrom P, Roberts J (1990) Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica 58:1255–1277

    Article  Google Scholar 

  • Milgrom P, Shannon C (1994) Monotone comparative statics. Econometrica 62:157–180

    Article  Google Scholar 

  • Monderer D, Shapley LS (1996) Potential Games. Games Econ Behav 14:124–143

    Article  Google Scholar 

  • Morris S, Rob R, Shin HS (1995) \(p\)-Dominance and belief potential. Econometrica 63:145–157

    Article  Google Scholar 

  • Morris S, Ui T (2005) Generalized potentials and robust sets of equilibria. J Econ Theory 124:45–78

    Article  Google Scholar 

  • Oyama D, Takahashi S (2011) On the relationship between robustness to incomplete information and noise-independent selection in global games. J Math Econ 47:683–688

    Article  Google Scholar 

  • Oyama D, Tercieux O (2009) Iterated potential and robustness of equilibria. J Econ Theory 144:1726–1769

    Article  Google Scholar 

  • Pesendorfer W (1995) Design innovations in fashion cycles. Am Econ Rev 85:771–792

    Google Scholar 

  • Plott CR, Smith J (1999) Instability of equilibria in experimental markets: upward-sloping demand externalities. South Econ J 65:405–426

    Article  Google Scholar 

  • Quint T, Shubik M (1997) A theorem on the number of nash equilibria in a Bimatrix game. Int J Game Theory 26:353–359

    Article  Google Scholar 

  • Quint T, Shubik M (2002) A bound on the number of nash equilibria in a coordination game. Econ Lett 77:323–327

    Article  Google Scholar 

  • Ritzberger K, Weibull JW (1995) Evolutionary selection in normal-form games. Econometrica 63:1371–1399

    Article  Google Scholar 

  • Rohlfs JH (2001) Bandwagon effects in high technology industries. MIT Press, Cambridge

    Google Scholar 

  • Savani R, von Stengel B (2006) Hard-to-solve Bimatrix games. Econometrica 74:397–429

    Article  Google Scholar 

  • Savani R, von Stengel B (2015) Game theory explorer: software for the applied game theorist. CMS 12:5–33

    Article  Google Scholar 

  • Shapley LS (1974) A note on the Lemke–Howson algorithm. Math Programm Stud 1:175–189

    Article  Google Scholar 

  • The Economist (2013) Browser wars: chrome rules the web. https://www.economist.com/news/business/21583288-what-googles-browser-has-common-queen-victoria-chrome-rules-web. Accessed 10 Aug 2017

  • Topkis DM (1998) Supermodularity and complementarity. Princeton University Press, Princeton

    Google Scholar 

  • Ui T (2001) Robust equilibria of potential games. Econometrica 69:1373–1380

    Article  Google Scholar 

  • Vives X (1990) Nash equilibrium with strategic complementarities. J Math Econ 19:305–321

    Article  Google Scholar 

  • Vives X (2001) Oligopoly pricing: old ideas and new tools. MIT Press, Cambridge

    Google Scholar 

  • von Stengel B (1999) New maximal numbers of equilibria in Bimatrix games. Discrete Comput Geom 21:557–568

    Article  Google Scholar 

  • Young P (1993) The evolution of conventions. Econometrica 61:57–84

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jun Honda.

Additional information

This subsumes the previous working paper: “Equilibrium selection for symmetric coordination games with an application to the minimum-effort game”. I am deeply indebted to Josef Hofbauer for his supervision and guidance throughout this project. I am also grateful to Satoru Takahashi and Bernhard von Stengel for insightful suggestions and helpful comments that substantially improved the paper. I would like to thank Ulrich Berger, Ezra Einy, Maarten Janssen, Michihiro Kandori, Wieland Müller, Daisuke Oyama, Christina Pawlowitsch, Motty Perry, Klaus Ritzberger, Dov Samet, Karl Schlag, Yasuhiro Shirata, Joel Sobel, Gerhard Sorger, Olivier Tercieux, Naoki Yoshihara, Boyu Zhang, and the audience at Games and Strategy in Paris: International Conference for Sylvain Sorin’s Sixties, the 23rd International Conference on Game Theory at Stony Brook University, Games 2012, EEA ESEM Málaga 2012, Micro Research Seminar of Vienna Graduate School of Economics for valuable comments and discussions. Finally, I am grateful to the co-editor, Vijay Krishna, and the two anonymous referees for their constructive comments, which helped enhance the quality of the paper and correct a critical part of my arguments.

Appendix: proofs and examples

Appendix: proofs and examples

Consider any given finite set of strategies A with \(|A|=n\). Any game \(\mathbf {g}=(A,g)\) with TBP is nondegenerate. From Shapley (1974, Theorem 2) and Quint and Shubik (1997, Lemma 2.2), we know that the number of NE is both finite and odd if a game is nondegenerate, and also that for any given subset \(S \subseteq A\) there exists at most one symmetric NE with support S, otherwise all convex combinations of two NE with identical support would constitute a continuum of NE, which violates the condition that the game has a finite number of NE. Therefore, game \(\mathbf {g}=(A,g)\) with TBP has at most symmetric \(2^{n}-1\) NE. To prove Proposition 1 and Theorem 1, we will apply these associated facts with TBP:

  1. (f1)

    the finiteness property (the number of NE is finite);

  2. (f2)

    the oddness property;

  3. (f3)

    the upper bound of the number of symmetric NE is \(2^{n}-1\).

1.1 Proof of Proposition 1

Proof

Consider any given symmetric game \(\mathbf {g}=(A,g)\) with TBP. Assume that the game has an asymmetric NE, \((x^{1},x^{2})\in \varDelta ^{2}\) with \(x^{1}\ne x^{2}\). There are two cases to consider: (i) \(\text {supp}(x^{1})=\text {supp}(x^{2})\) and (ii) \(\text {supp}(x^{1})\ne \text {supp}(x^{2})\). We show below that neither case can hold.

Consider case (i) and take \(S = \text {supp}(x^{1})=\text {supp}(x^{2}) \subseteq A\) for \(x^{1}\ne x^{2}\). Since \((x^{1},x^{2})\in \varDelta ^{2}\) is in equilibrium, \(\text {br}(x^{1})=\text {br}(x^{2}) = S\). With symmetry, \((x^{2},x^{1})\in \varDelta ^{2}\) is also an equilibrium with support S. Let \(\hat{\varDelta }(S) \equiv \{ x \in A \mid \forall \lambda \in [0,1], x = \lambda x^1 + (1-\lambda )x^2 \}\) be the set of all convex combinations of two points \(x^1\) and \(x^2\). Since the set of best responses is convex, \(\text {br}(x) = S\) must hold for all \(x \in \hat{\varDelta }(S)\), implying that any pair of two points in the line segment of \(\hat{\varDelta }(S)\) constitutes a continuum of NE. Since the game with TBP is nondegenerate, though, the number of NE must be finite due to the associated fact (f1) with TBP, which contradicts the existence of the continuum of NE.

In case (ii), without loss of generality, there must be some pure strategy \(k \in \text {supp}(x^{1}) \backslash \text {supp}(x^{2})\). By TBP, player 1 takes a non-best response k to the opponent’s strategy \(x^{2}\), which is a contradiction because all pure strategies in \(\text {supp}(x^{1})\) must be the best response to \(x^{2}\) in equilibrium.

1.2 Proof of Theorem 1

We will prove the if part and the only if part of Theorem 1 in order. For the only if part, we will use the associated facts (f2) and (f3) with TBP mentioned above.

1.2.1 Proof of the if part

To begin with, I first provide a lemma, which is subsequently used in the proof of the if part of Theorem 1. For this, consider a symmetric game \(\mathbf {g}=(A,g)\) with \(|A| = n\). Suppose that \(\mathbf {g}\) has exactly \(2^{n}-1\) symmetric NE. For any given subset of strategies \(S\subseteq A\) we denote by \(x^{*}|_{S} \in \varDelta \) the strategy in the symmetric NE with support S. Those NE must have different supports from one another, that is, they must be regular (Harsanyi 1973), and include all symmetric pure strategy profiles, otherwise there are at least two distinct symmetric (mixed strategy) NE with identical support and all convex combinations of them will be NE, and so the number of NE would be infinite, leading to a contradiction. By the same token, we can show:

Lemma 2

Suppose that a symmetric game \(\mathbf {g}=(A,g)\) with \(|A| = n\) has \(2^n -1\) symmetric Nash equilibria. For any given \(S \subseteq A\), \(x^{*}|_{S}\) satisfies

$$\begin{aligned} \text {br}(x^{*}|_{S})= & {} \text {supp}(x^{*}|_{S}). \end{aligned}$$
(A.1)

Proof

By assumption, there are \(2^n -1\) symmetric NE for all distinct supports \(S \subseteq A\). Take one of them for any given \(S \subseteq A\). Denote its equilibrium strategy by \(x^{*}|_{S} \in \mathring{\varDelta }(S)\). By the property of NE, \( \text {supp}(x^{*}|_{S}) \subseteq \text {br}(x^{*}|_{S})\) holds true. Below we show that \(\text {supp}(x^{*}|_{S}) \) cannot be a proper subset of \( \text {br}(x^{*}|_{S})\), and therefore (A.1) must hold instead.

Consider first the case of \(S = A\). In this case, \(\text {br}(x^{*}|_{S}) \subseteq A = \text {supp}(x^{*}|_{S})\) holds. Together with \( \text {supp}(x^{*}|_{S}) \subseteq \text {br}(x^{*}|_{S})\), this implies that (A.1) must hold.

Next consider any given \(S \subsetneq A\) with \(|S| =1, \dots ,n-1\) and the respective symmetric NE strategy of \(x^{*}|_{S}\).

For this, suppose that there exists a pure strategy \(j \in A\) such that \(j \in \text {br}(x^{*}|_{S}) \backslash \text {supp}(x^{*}|_{S})\). Let \(\tilde{S} = S \cup \{j\} \subseteq A\). By assumption, we know that there is a symmetric NE strategy, \(x^{*}|_{\tilde{S}}\), such that \(\tilde{S} \subseteq \text {br}(x^{*}|_{\tilde{S}})\).

With this, consider now all convex combinaitons connecting \(x^{*}|_{S}\) to \(x^{*}|_{\tilde{S}}\), and denote them by

$$\begin{aligned} X= & {} \{ x \in \mathring{\varDelta }(\tilde{S}) \mid \forall t \in [0,1], x = t x^{*}|_{S} + (1 - t) x^{*}|_{\tilde{S}} \}. \end{aligned}$$

Taking any point \(x \in X\) as given, \(\tilde{S} \subseteq \text {br}(x)\) must hold as x is a convex combination of two strategies to which \(\tilde{S}\) is the set of best responses. This implies that there is a continuum of equilibria along with X, which contradicts the fact that the game has a finite number of NE.

Thus, for any given \(S \subseteq A\), the symmetric NE strategy of \(x^{*}|_{S}\) satisfies (A.1).

With this lemma, we show the if part of Theorem 1 as follows.

Proof

Consider a symmetric game \(\mathbf {g}=(A,g)\) with \(|A| = n\) where there are \(2^{n}-1\) symmetric NE.

In the case of \(|\text {supp}(x)| = 1\), the strategy \(x \in \varDelta \) represents a pure strategy \(i \in A\). Since any symmetric pure strategy profile is a NE, Lemma 2 implies that \(\text {br}(x) = \text {supp}(x)\) must hold.

In the case of \(|\text {supp}(x)| = n\), both \(\text {br}(x) \subseteq A\) and \(\text {supp}(x) = A\) hold true, which leads to \(\text {br}(x) \subseteq \text {supp}(x)\).

So, what we have to show is that \(\text {br}(x) \subseteq \text {supp}(x)\) holds for any other \(x \in \varDelta \) with \(|\text {supp}(x)| \in \{2, \dots , n-1\}\) when \(n \ge 3\).

Suppose \(n \ge 3\). For any given \(x \in \varDelta \) with \(|\text {supp}(x)| \in \{2,\dots , n-1\}\), consider the sequence of tuples composed of the subsets of strategies and their associated subsets and constants \( (S^k, T^k,c^k)_{k=0}^{m}\) with \(S^k \subseteq A\), \(T^k \subseteq S^k\), and \(c^k>0\) for some \(m \in \mathbb {N}_{+}\) and \(k=0,\dots ,m\) such that

$$\begin{aligned}&S^0 = \text {supp}(x), \quad T^0 = \arg \min _{i \in S^0} \frac{x_i}{x_i^{*}|_{S^0}}, \quad c^0 = \min _{i \in S^0} \frac{x_i}{x_i^{*}|_{S^0}}, \\&S^{l+1} = S^l \backslash T^l, \quad T^{l+1} = \arg \min _{i \in S^{l+1}} \frac{x_i - \sum _{j< l}c^j x_{i}^{*}|_{S^j}}{x_i^{*}|_{S^{l+1}}}, \\&c^{l+1} = \min _{i \in S^{l+1}} \frac{x_i - \sum _{j < l}c^j x_{i}^{*}|_{S^j}}{x_i^{*}|_{S^{l+1}}} \end{aligned}$$

for \(l = 0,\dots ,m-1\) and \(\emptyset = S^{m+1} \subsetneq S^m \subsetneq \cdots \subsetneq S^1 \subsetneq S^0 \subseteq A\). With this sequence, we seek to exclude all non-best responses in \(\text {supp}(x)\) on the basis of the equilibria step-by-step so as to extract the best responses from \(\text {supp}(x)\).

We can now decompose x into \((m+1)\) parts as

$$\begin{aligned} x= & {} \sum _{k=0}^m c^k x^{*}|_{S^k} \end{aligned}$$

where \(x^*|_{S^k}\) satisfies, by Lemma 2, \(\text {br}(x^*|_{S^k}) = \text {supp}(x^*|_{S^k}) = S^k\) for any given \(k = 0,\dots ,m\). Let

$$\begin{aligned} \pi _i^k= & {} \sum _{j \in S^k} g_{ij} x_j^{*}|_{S^k} \end{aligned}$$

be the expected payoff of strategy i given that the opponent plays \(x^{*}|_{S^k}\) for any given \(k = 0,\dots ,m\). With this, the condition of \(\text {br}(x^*|_{S^k}) = \text {supp}(x^*|_{S^k}) = S^k\) implies that the strict inequality of \(\pi _i^k > \pi _h^k\) holds true for all \(i \in S^k\) and \(h \in A \backslash S^k\), whereas the equality of \(\pi _i^k = \pi _h^k\) for any given \(i,h \in S^k\). Furthermore, let

$$\begin{aligned} \pi _i(x)= & {} \sum _{k = 0}^m c^k \pi _i^k \end{aligned}$$

be the payoff of strategy i given that the opponent plays x. By construction, \(\pi _i(x) > \pi _h(x)\) holds for all \(i \in S^m\) and \(h \in A \backslash S^m\) while \(\pi _i(x) = \pi _h(x)\) for any given \(i,h \in S^m\), thus \(\text {br}(x) = S^m\) holds true. Together with \(S^m \subseteq S^0 = \text {supp}(x)\), this leads to \(\text {br}(x) \subseteq \text {supp}(x)\).

From above, we have shown that \(\text {br}(x) \subseteq \text {supp}(x)\) holds for all \(x \in \varDelta \).

1.2.2 Proof of the only if part

We show the only if part of Theorem 1 by induction. In doing so, we use Proposition 1 to ensure that there are no asymmetric NE in any given symmetric game with TBP, by which we can restrict attention to symmetric NE. For any given two-player symmetric game \({\mathbf{g}}=(A,g)\) and any given subset \(S \subset A\), define by \({\mathbf{g}}|_{S}\) a restricted game in which both players choose their strategies only from S.

First, we show:

Lemma 3

Let \({\mathbf{g}}=(A,g)\) be any symmetric \(n \times n\) game with TBP for any integer \(n \ge 2\). Then any restricted game \({\mathbf{g}}|_{S}\) with \(|S|\le k(=2,3,\dots ,n)\) has a unique symmetric Nash equilibrium with support S.

Proof

We show Lemma 3 by induction as follows.

  1. (I)

    It is obvious for \(k=2\).

  2. (II)

    Suppose that every restricted game \({\mathbf{g}}|_{S}\) with the support size |S| that is smaller than or equal to \(k(=2,3,\dots ,n-1)\) has a unique symmetric NE with support S. Taking into account that any restricted game \({\mathbf{g}}|_{S}\) has TBP if \({\mathbf{g}}\) has TBP, this implies that the restricted game \({\mathbf{g}}|_{S^{\prime }}\) with \(|S^{\prime }|=k^{\prime }(=1,\dots ,k)\) has \(2^{k^{\prime }}-1\) symmetric NE. Take now any (restricted) game \({\mathbf{g}}|_{S^{\prime \prime }}\) with \(|S^{\prime \prime }|=k+1\), and consider all restricted games of \({\mathbf{g}}|_{S^{\prime \prime }}\). Each of those restricted games has at least one strategy in \(S^{\prime \prime }\) and at most \(k+1\) strategies. The number of those restricted games except for \({\mathbf{g}}|_{S^{\prime \prime }}\) is \(2^{k+1}-2\). Again by TBP, \({\mathbf{g}}|_{S^{\prime \prime }}\) should have at least \(2^{k+1}-2\) symmetric NE. Since any game with TBP is nondegenate, using the fact (f3) we know that \({\mathbf{g}}|_{S^{\prime \prime }}\) has at most \(2^{k+1}-1\) symmetric NE. Together with the fact (f2), \({\mathbf{g}}|_{S^{\prime \prime }}\) must have a unique NE with support \(S^{\prime \prime }\), and this NE must be symmetric due to Proposition 1.

Using Lemma 3, we can show the only if part of Theorem 1.

Proof

Any restricted game \({\mathbf{g}}|_{S}\) with \(|S|\le n-1\) has a unique symmetric NE with support S due to Lemma 3, and the number of all restricted games \({\mathbf{g}}|_{S}\) with \(|S|\le n-1\) is \(2^{n}-2\). Since any restricted game \({\mathbf{g}}|_{S}\) has TBP if \({\mathbf{g}}\) has TBP, the game \({\mathbf{g}}=(A,g)\) has at least \(2^{n}-2\) symmetric NE, which do not include an interior NE in \(\varDelta \). Following Lemma 3, \({\mathbf{g}}\) must have a unique interior (symmetric) NE and therefore it has \((2^{n}-2)+1=2^{n}-1\) symmetric NE.

1.3 Proof of Theorem 2

1.3.1 Proof of the if part

Proof

Fix any given \(i,j=1,2\) with \(i \ne j\) and \(x \in \varDelta \). In the same way as in the proof of the if part of Theorem 1, we can show that \(\text {br}_i(x) \subseteq \text {supp}(x)\) holds for player i by constructing a sequence of tuples composed of subsets of strategies and their associated subsets and constants on the basis of player j’s equilibrium strategies that have different supports from one another.

1.3.2 Proof of the only if part

The main point of the proof is to use each player’s payoff function for finding out a symmetric NE in a “fictitious” symmetric game, which gives two (possibly different) symmetric NE with the identical support, and to combine those two NE strategies to construct a NE in the original game.

Proof

Fix any game \({\mathbf{g}}=(\{1,2\},A,(g^{i})_{i=1,2})\) where \(A=\{1,2,\dots ,n\}\) and GTBP holds. For any given player \(i=1,2\), construct the symmetric two-player game by using the player i’s payoff function \(g^{i}\). Denote it by \({\mathbf{g}}^{i}=(A,g^{i})\). Since \({\mathbf{g}}^{i}\) has the same set of strategies A for both two players and the symmetric payoff function \(g^{i}\), the game \({\mathbf{g}}^{i}\) satisfies TBP by GTBP. From Theorem 1, if \({\mathbf{g}}^{i}\) has TBP, it has \(2^{n}-1\) symmetric NE that are all different in terms of supports. Consider the symmetric NE \((x^{i}|_{S},x^{i}|_{S})\in \varDelta \) with \(\text {supp}(x^{i}|_{S})=S \subseteq A\) in \({\mathbf{g}}^{i}\). The strategy profile \((x^{2}|_{S},x^{1}|_{S})\) is a NE of the game \({\mathbf{g}}\) such that \(\text {supp}(x^{1}|_{S})=\text {supp}(x^{2}|_{S})=S\). Since \({\mathbf{g}}^{i}\) has \(2^{n}-1\) symmetric NE for every \(i=1,2\), the bimatrix game \({\mathbf{g}}=(\{1,2\},A,(g^{i})_{i=1,2})\) also has \(2^{n}-1\) NE, and any two of those NE are described by \((x^{2}|_{S},x^{1}|_{S})\) and \((x^{2}|_{S^{\prime }},x^{1}|_{S^{\prime }})\) for \(S,S^{\prime } \subseteq A\) with \(S \ne S^{\prime }\), respectively.

1.3.3 Examples

In the following, we give an example regarding Theorem 2.

Example 1

(Asymmetric pure-coordination game) Let us consider the \(4\times 4\) pure-coordination game where \(A=\{1,2,3,4\}\) and for any player \(i=1,2\), \(g_{kh}^i>0\) for \(k=h\), otherwise \(g_{kh}^i=0\). With player i’s payoffs, we define 11 different mixed strategies \(x^j \in \varDelta \) of player \(j \ne i\) based on these support sizes |S|: 6 different strategies in the case of \(|S|=2\), for any two different strategies \(h,l \in A\) and any other strategy \(k \in A\backslash \{h,l\}\),

$$\begin{aligned} (x_h^j,x_l^j) =\frac{1}{g_{hh}^i+g_{ll}^i}(g_{ll}^i ,g_{hh}^i), \quad x_k^j=0; \end{aligned}$$

4 different strategies in the case of \(|S|=3\), for any three different strategies \(h,l,k \in A\) and any other strategy \(m \in A\backslash \{h,l,k\}\),

$$\begin{aligned} (x_h,x_l,x_k) = \frac{1}{g_{hh}^i g_{ll}^i + g_{hh}^i g_{kk}^i+g_{ll}^i g_{kk}^i } (g_{ll}^ig_{kk}^i,g_{hh}^ig_{kk}^i,g_{hh}^ig_{ll}^i), \quad x_m=0; \end{aligned}$$

1 strategy in the case of \(|S|=4\),

$$\begin{aligned}&\frac{1}{g_{11}^i g_{22}^i g_{33}^i + g_{11}^i g_{22}^i g_{44}^i + g_{11}^i g_{33}^i g_{44}^i+ g_{22}^i g_{33}^i g_{44}^i }\\&\quad (g_{22}^ig_{33}^i g_{44}^i, g_{11}^i g_{33}^i g_{44}^i, g_{11}^i g_{22}^i g_{44}^i, g_{22}^i g_{33}^i g_{44}^i). \end{aligned}$$

The game has \(2^{4}-1(=15)\) NE consisting of the pair of the eleven mixed strategies described above and also of the four symmetric pure-strategy profiles. One can easily observe that the game satisfies GTBP.

To illustrate how the proof of the if part of Theorem 2 works out, consider, for instance, the payoffs of \(g_{11}^1=g_{44}^2=1\), \(g_{22}^1=g_{33}^2=2\), \(g_{33}^1=g_{22}^2=3\), \(g_{44}^1=g_{11}^2=4\), and then take \(i=1\) and a mixed strategy \(x =(4/7 , 2/7 ,1/7 , 0) \in \varDelta \) in order to demonstrate that \(\text {br}_1(x) \subseteq \text {supp}(x)\) holds. From above, we can easily compute all NE of the game. Denote by \(x^{*}|_S =(x_h^{*}|_S)_{h=1}^4 \in \varDelta \) the player 2’s strategy in the NE with support \(S \subseteq A\). With these equilibrium strategies of player 2, we construct the sequence of tuples composed of subsets of strategies and their associated subsets and constants by \((S^k,T^k,c^k)_{k}\) such that

$$\begin{aligned} S^0= & {} \text {supp}(x) = \{1,2,3\}, \quad T^0 = \arg \min _{h \in S^0} \frac{x_h}{x_h^{*}|_{S^0}} = \{ 3\}, \\ c^0= & {} \min _{h \in S^0} \frac{x_h}{x_h^{*}|_{S^0}} = \frac{1/7}{ 2/11} = \frac{11}{14}, \\ S^1= & {} S^0 \backslash T^0 = \{1,2\}, \quad T^1 = \arg \min _{h \in S^1} \frac{x_h - c^0 x_{h}^{*}|_{S^0}}{x_h^{*}|_{S^1}} = \{1,2\}, \\ c^1= & {} \min _{h \in S^1} \frac{x_h - c^0 x_{h}^{*}|_{S^0}}{x_h^{*}|_{S^1}} = \left. \frac{4/7 - (11/14)(6/11)}{2/3}\right| _{h=1} \\= & {} \left. \frac{2/7 - (11/14)(3/11)}{1/3}\right| _{h=2} = \frac{3}{14}, \end{aligned}$$

and \(S^{2} = S^1 \backslash T^1 = \emptyset \), where \(x^{*}|_{S^0} = (6/11,3/11,2/11,0)\) and \(x^{*}|_{S^1} = (2/3,1/3,0,0)\). This sequence leads to \(\text {br}_1(x) \subseteq \text {supp}(x)\), because \(\text {br}_1(x) = \text {br}_1(x^*|_{S^0}) \cap \text {br}_1(x^*|_{S^1}) = \{1,2\} = \text {supp}(x^*|_{S^0}) \cap \text {supp}(x^*|_{S^1}) = S^0 \cap S^1 \subseteq \{1,2,3\}= S^0 = \text {supp}(x) \).

Note that GTBP holds for all \(n\times n\) pure coordination games and those slightly perturbed ones (see footnote 9 for the definition of a slightly perturbed pure coordination game).

1.3.4 Games with \(2^{n}-1\) NE without the restriction on the support of NE

GTBP does not necessarily hold for a given asymmetric game \({\mathbf{g}}\) with \(2^{n}-1\) NE. To ensure GTBP, \({\mathbf{g}}\) must be nondegenerate and also satisfy the restriction on the support of NE imposed in Theorem 2. Below we will provide those corresponding counterexamples.

Example 2

(Degenerate \(3\times 3\) coordination game) Consider the asymmetric \(3\times 3\) coordination game given by Table 1. The game has 3 symmetric pure-strategy NE as well as 4 other mixed-strategy NE:

$$\begin{aligned}&((2/3,1/3,0),(1/2,1/2,0)),((1/2,0,1/2),(1/2,0,1/2)),\\&\quad ((0,3/5,2/5),(0,1/2,1/2)),((2/7,3/7,2/7),(1/2,0,1/2)). \end{aligned}$$

While the game has \(7(=2^{3}-1)\) NE, the strategy profile \((x^{1},x^{2})=((2/7,3/7,2/7),(1/2,0,1/2))\) violates GTBP as the game is degenerate with \(|\text {br}_{1}(x^{2})|>|\text {supp}(x^{2})|\).

Table 1 An asymmetric \(3\times 3\) coordination game

Example 3

(Nondegenerate asymmetric \(4\times 4\) game) Consider the asymmetric \(4\times 4\) game given by Table 2.Footnote 17 This game has the following 15 NE.Footnote 18 \(|\{(x^{1},x^{2})\in \text {NE}\mid \forall i=1,2,|\text {supp}(x^{i})|=1\}|=2:\)

$$\begin{aligned} \left( (0,1,0,0),(0,0,0,1)),((0,0,1,0),(1,0,0,0) \right) . \end{aligned}$$

\(|\{(x^{1},x^{2})\in \text {NE}\mid \forall i=1,2,|\text {supp}(x^{i})|=2\}|=10:\)

$$\begin{aligned}&\left( (11/15,4/15,0,0),(4/15,11/15,0,0) \right) , \end{aligned}$$
(A.2)
$$\begin{aligned}&\left( (0,0,1/2,1/2),(0,0,1/2,1/2) \right) ,\\&\left( (51/70,19/70,0,0),(0,23/27,4/27,0) \right) , \nonumber \\&\left( (23/27,0,0,4/27) (19/70,51/70,0,0) \right) ,\nonumber \\&\left( (2/7,5/7,0,0),(0,0,19/42,23/42) \right) , \nonumber \\&\left( (93/112,0,0,19/112),(0,93/112,19/112,0) \right) ,\nonumber \\&\left( (0,0,23/42,19/42),(2/7,5/7,0,0) \right) , \nonumber \\&\left( (0,0,31/59,28/59),(0,15/19,4/19,0) \right) ,\nonumber \\&\left( (15/19,0,0,4/19),(0,0,28/59,31/59) \right) , \nonumber \\&\left( (0,33/53,20/53,0),(33/53,0,0,20/53)\right) .\nonumber \end{aligned}$$
(A.3)

\(|\{(x^{1},x^{2})\in \text {NE}\mid \forall i=1,2,|\text {supp}(x^{i})|=3\}|=2:\)

$$\begin{aligned}&\left( (60/109,39/109,10/109,0),(31/74,0,17/111,95/222) \right) ,\\&\left( (0,31/74,95/222,17/111),(39/109,60/109,0,10/109) \right) . \end{aligned}$$

\(|\{(x^{1},x^{2})\in \text {NE}\mid \forall i=1,2,|\text {supp}(x^{i})|=4\}|=1:\)

$$\begin{aligned} \left( (5/11,4/11,5/33,1/33),(4/11,5/11,1/33,5/33) \right) . \end{aligned}$$
(A.4)

The game is nondegenerate and has \(15(=2^{4}-1)\) NE but does not satisfy GTBP because it is not a coordination game (even if we can allow for all kinds of permutations of the strategies). The condition that both players have an identical support in every NE holds only for 3 NE of (A.2)–(A.4).

Table 2 An asymmetric \(4\times 4\) game

Example 3 illustrates that a nondegenerate \(n \times n\) game with \(2^{n}-1\) NE does not need to be a coordination game. To show that a nongedenerate \(n \times n\) game with \(2^{n}-1\) NE has GTBP, the game must be a coordination game, but this alone does not suffice. Especially when there are many actions, we must impose the restriction on the support of NE.

1.4 Proof of Lemma 1

Proof

Suppose that a game with TBP has a GPRD–equilibrium \((i,i)\in A^2\). Then, br\((x) = \{i \}\) must hold for \(x \in \mathring{\varDelta }(\{i,j\})\) with \(x_i=x_j=1/2\) for any given pure strategy \(j \in A\backslash \{i\}\). Since all pure-strategy best-response sets are convex in \(\varDelta \), this implies that br\((x) = \{i \}\) holds for all \(x \in \varDelta \) with \(x_i \ge 1/2\), thus (ii) is a 1 / 2–dominant equilibrium.

In the following we provide an example to highlight the difference between Lemma 1 in this paper and Lemma 2 in Kandori and Rob (1998).

Example 4

(Symmetric \(3\times 3\) game with/out TBP) Kandori and Rob (1998, p.44–45) consider a symmetric \(3\times 3\) game as shown in the left panel of Table 3, where only the row player’s payoffs are given due to symmetry. The associated best response region is depicted as the left panel of Fig. 1, where the numbers in parentheses represent the best response for each region. The figure indicates that TBP fails as there are mixed strategies \(x \in \varDelta (\{1,3\})\) such that \(\text {br}(x) = 2 \notin \{1,3\} = \text {supp}(x)\), and also that the strategy profile (3, 3) is a GPRD-equilibrium as well as a 1 / 2-dominant equilibrium. From Proposition 2 and the subsequent argument for Figures 1–3 in Kandori and Rob (1998), MBP is shown to hold in this game.

Table 3 A symmetric \(3 \times 3\) game where MBP holds but TBP fails (left panel); TBP holds but MBP fails (right panel)
Fig. 1
figure 1

The best response regions of the games on the left-hand-side (left panel) and on the right-hand-side (right panel) in Table 3

Consider now a symmetric \(3\times 3\) game as shown in the right panel of Table 3. The associated best response region is described as the right panel of Fig. 1. The figure indicates that TBP holds true as well as (3, 3) is a 1 / 2-dominant equilibrium, while showing that MBP fails as the borderline of the best response regions of strategies 1 and 2 should lie in the gray-colored region to meet MBP (see Kandori and Rob 1998, for the detail).

If a symmetric \(3 \times 3\) game satisfies both TBP and \(g_{23} - g_{13} > g_{22} - g_{12}\), MBP should fail (Kandori and Rob 1998, Proposition 2). The game shown in the right panel of Table 3 is supermodular and meets both, therefore violating MBP.

1.5 Proof of Proposition 2

We first introduce the following notations. For \(x,x^{\prime }\in \varDelta \), we write \(x\succ x^{\prime }\) if x stochastically dominates \(x^{\prime }\), that is, for any \( i \in A\), \(\sum _{i \le j \le n}x_{j} \ge \sum _{i \le j \le n} x_{j}^{\prime }\) holds with strict inequality for at least some i. In particular when considering a mixed strategy \(x \in \mathring{\varDelta }(\{i,j\})\) for \(i,j\in A\) with \(i \ne j\), we write \(x^{i j}\) instead of x for clarity. Let \(\min \text {br}(x)\) be the lowest pure-strategy best-response against the opponent’s strategy x and \(\text {max}\text {br}(x)\) the highest. With these notations, we prove Proposition 2 by using the property on supermodular games \({\mathbf{g}}=(A,g)\) that the best-response correspondence for any given player is non-decreasing in the opponent’s strategies, that is, for any \(x,x^{\prime }\in \varDelta \) with \(x\succ x^{\prime }\), both \(\min \text {br}(x)\ge \min \text {br}(x^{\prime })\) and \(\max \text {br}(x)\ge \max \text {br}(x^{\prime })\) hold.Footnote 19

Proof

By TBP, it follows that for \(x^{1n} \in \mathring{\varDelta }(\{1,n\})\) with \(x_{1}^{1n}=x_{n}^{1n}=1/2\),

$$\begin{aligned} \text {br}(x^{1n})\subseteq \{1,n\}. \end{aligned}$$

By the assumption of \(g_{11}-g_{n1}\ne g_{nn}-g_{1n}\), . Suppose that holds, so does . Now take with . Since \(x^{1n}\succ x^{1i}\) holds for , by supermodularity, it follows that

$$\begin{aligned} \{1\}=\max \text {br}(x^{1n})\ge \max \text {br}(x^{1i}), \end{aligned}$$

which implies that

$$\begin{aligned} \text {br}(x^{1i})&=\{1\} \end{aligned}$$

holds for all \(i \in A\) and the belief \(x^{1i} \in \mathring{\varDelta }(\{1,i\})\) with \(x_{1}^{1i} = x_{i}^{1i}=1/2\). This condition implies that the strategy profile (1,1) is a GPRD-equilibrium, and by Lemma 1, it is a 1 / 2–dominant equilibrium.

To show the uniqueness, suppose that there are two 1 / 2–dominant equilibria, (ii) and (jj). This gives the following two inequalities,

$$\begin{aligned} \dfrac{1}{2}g_{ii}+\dfrac{1}{2}g_{ij}>\dfrac{1}{2}g_{jj}+\dfrac{1}{2}g_{ji},\\ \dfrac{1}{2}g_{jj}+\dfrac{1}{2}g_{ji}>\dfrac{1}{2}g_{ii}+\dfrac{1}{2}g_{ij}, \end{aligned}$$

which leads to a contradiction.

Similarly, in the case of \(g_{11}-g_{n1} < g_{nn}-g_{1n}\), we can show that (nn) is the unique 1 / 2–dominant equilibrium.

1.6 Proof of Proposition 3

Proof

Suppose that there is a potential function v such that (ii) is a unique potential maximizer. Then, both \(v_{ii}>v_{jj}\) and \(v_{ij}=v_{ji}\) must hold for any other strategy \(j \in A \backslash \{ i\}\), which leads to

$$\begin{aligned} v_{ii} - v_{jj} > 0 = v_{ji} - v_{ij}. \nonumber \end{aligned}$$

Rewrite this as \(v_{ii} - v_{ji} > v_{jj} - v_{ij}\). By definition of the potential maximizer, \(g_{ii} - g_{ji} > g_{jj} - g_{ij}\) should hold, which implies that

$$\begin{aligned} \frac{1}{2}( g_{ii} + g_{ij} ) > \frac{1}{2} ( g_{ji} + g_{jj} ) \nonumber \end{aligned}$$

holds for all \(j\in A \backslash \{i\}\), and therefore (ii) is a GPRD–equilibrium. Together with Lemma 1, it is a 1 / 2–dominant equilibrium.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Honda, J. Games with the total bandwagon property meet the Quint–Shubik conjecture. Int J Game Theory 47, 893–912 (2018). https://doi.org/10.1007/s00182-017-0609-3

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00182-017-0609-3

Keywords

JEL Classification

Navigation