Abstract
In this paper, we study games where the space of players (or types, if the game is one of incomplete information) is atomless and payoff functions satisfy the property of strict single crossing in players (types) and actions. Under an additional assumption of quasisupermodularity in actions of payoff functions and mild assumptions on the player (type) space—partially ordered and with sets of uncomparable players (types) having negligible size—and on the action space—lattice, second countable and satisfying a separation property with respect to the ordering of actions—we prove that every Nash equilibrium is essentially strict. Furthermore, we show how our result can be applied to incomplete information games, obtaining the existence of an evolutionary stable strategy, and to population games with heterogeneous players.
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Notes
Second countability implies a cardinality less than or equal to the cardinality of the continuum.
The separation property that we assume ensures that every two actions that can be strictly ordered can also be separated by a third action not greater than the largest of the two.
Here we follow the labeling proposed by Khan and Sun (2002), which allows to encompass both games with many players and games with incomplete information.
For games with incomplete information, the set I of groups/institutions has to be interpreted as the set of players, while the set of players \(T_i\) has to be interpreted as the set of types for player \(i \in I\).
We note that, under this definition of F as uncountable cross product of action sets, measurability issues can emerge. These issues cannot be settled without imposing further structure, that is however unnecessary for our main result. Therefore, we choose to take care of measurability only in the applications of Sect. 4.
In case of incomplete information games (where i is intepreted as a player and \(T_i\) as her set of types), player i has already known her type t when computing expected utility. So, it is redundant to consider the actions that would be taken by types in \(T_i {\setminus } \{t\}\), and hence, we have to require that \(u_i(t,f)\) is constant over the actions chosen by types \(t' \in T_i{\setminus } \{t\}\).
T0 requires that any two distinct points in a set are topologically distinguishable, i.e., the sets of neighborhoods of the two points differ one from the other.
We note that AU2 is slightly different from the standard definition of strict single crossing property since the profile of opponents’ actions, which is a third argument of function u in addition to t and \(f_t\), is not exactly the same in \(f_{-t}\) and \(f_{-t'}\): Indeed, the behavior of players different from t and \(t'\) is the same, while the behavior of t is considered in \(f_{-t}\) but not in \(f_{-t'}\), and the behavior of \(t'\) is considered in \(f_{-t'}\) but not in \(f_{-t}\). This difference disappears if, for instance, we assume individual negligibility (see discussion at the end of Sect. 3) or if we constrain players to care only about actions of groups/institutions different from theirs (as it happens, e.g., in games with incomplete information).
Note that strict Nash equilibria are called strong Nash equilibria in Harsanyi (1973).
See also Dubey et al. (1980) for a related use of strict equilibria in large games.
Interest in games with many players has recently spanned across different settings (see, e.g., Alós-Ferrer and Ritzberger 2013, for extensive form games and Balbus et al. 2013, for games with differential information), and different notions of equilibrium (see, e.g., Correa and Torres-Martínez 2014, can exists when the make for essential equilibria).
See Carmona and Podczeck (2009) for a discussion on the relationship between alternative formalizations of non-atomic games and existence results, with a focus on large games. See also Fu and Yu (2015) for a discussion of the connection between the class of large games and the class of finite-player Bayesian games.
McAdams (2006) applies and extends this setup to prove existence of pure Nash equilibria in multiunit auctions.
As noted by McAdams (2003), the assumptions of a common support for types and a common set for actions are just for notational simplicity and can be safely removed.
This last assumption can be easily generalized to any form of trait aggregation, in the same way as it is typically done for aggregative games (see, e.g., Acemoglu and Jensen 2013).
We also note that the existence of a Nash equilibrium is not an issue in this game, e.g., one can invoke Theorem 1, point (i), in Khan et al. (2013a).
Even if we have not found a precise reference, it follows almost directly from the Proof of Theorem 1 in McAdams (2003) that, if we restrict attention to symmetric profiles in a symmetric game, then we are still able to show existence of an isotone pure-strategy equilibrium, which is hence symmetric.
References
Acemoglu, D., Jensen, M.K.: Aggregate comparative statics. Games Econ. Behav. 81, 27–49 (2013)
Alós-Ferrer, C., Ritzberger, K.: Large extensive form games. Econ. Theor. 52(1), 75–102 (2013)
Athey, S.: Single crossing properties and the existence of pure strategy equilibria in games of incomplete information. Econometrica 69, 861–889 (2001)
Balbus, Ł., Dziewulski, P., Reffett, K., Woźny, Ł.: Differential information in large games with strategic complementarities. Econ. Theory 59(1), 201–243 (2013)
Bomze, I., Pötscher, B.: Game Theoretic Foundations of Evolutionary Stability. Springer, Berlin (1989)
Carmona, G.: On the purification of Nash equilibria of large games. Econ. Lett. 85, 215–219 (2004)
Carmona, G.: Purification of Bayesian–Nash equilibria in large games with compact type and action spaces. J. Math. Econ. 44, 1302–1311 (2008)
Carmona, G., Podczeck, K.: On the existence of pure-strategy equilibria in large games. J. Econ. Theory 144, 1300–1319 (2009)
Correa, S., Torres-Martínez, J.P.: Essential equilibria of large generalized games. Econ. Theory 57(3), 479–513 (2014)
Crawford, V.P.: Nash equilibrium and evolutionary stability in large- and finite-population ‘playing the field’ models. J. Theor. Biol. 145, 83–94 (1990)
Dubey, P., Mas-Colell, A., Shubik, M.: Efficiency properties of strategies market games: an axiomatic approach. J. Econ. Theory 22(2), 339–362 (1980)
Dvoretzky, A., Wald, A., Wolfowitz, J.: Elimination of randomization in certain statistical decision procedures and zero-sum two-person games. Ann. Math. Stat. 22(1), 1–21 (1951)
Ely, J.C., Sandholm, W.: Evolution in Bayesian games I: theory. Games Econ. Behav. 53, 83–109 (2005)
Fu, H., Yu, H.: Pareto-undominated and socially-maximal equilibria in non-atomic games. J. Math. Econ. 58, 7–15 (2015)
Harsanyi, J.C.: Games with randomly disturbed payoffs. Int. J. Game Theory 2, 1–23 (1973)
Keisler, H.J., Sun, Y.: Why saturated probability spaces are necessary. Adv. Math. 221(5), 1584–1607 (2009)
Khan, M.A., Rath, K.P., Sun, Y.: The Dvoretzky–Wald–Wolfowitz theorem and purification in atomless finite-action games. Int. J. Game Theory 34(1), 91–104 (2006)
Khan, M.A., Rath, K.P., Sun, Y., Yu, H.: Large games with a bio-social typology. Journal of Economic Theory (2013a)
Khan, M.A., Rath, K.P., Sun, Y., Yu, H.: Strategic uncertainty and the ex-post nash property in large games. Theoretical Economics 10(1), 103–129 (2015)
Khan, M.A., Rath, K.P., Yu, H., Zhang, Y.: Large distributional games with traits. Econ. Lett. 118(3), 502–505 (2013b)
Khan, M.A., Sun, Y.: Pure strategies in games with private information. J. Math. Econ. 24(7), 633–653 (1995)
Khan, M.A., Sun, Y.: Non-cooperative games with many players. In: Aumann, R.J., Hart, S. (eds.) Handbook of Game Theory with Economic Applications, vol. 3, pp. 1761–1808. Elsevier, Amsterdam (2002). chapter 46
Mas-Colell, A.: On a theorem of Schmeidler. J. Math. Econ. 13, 206–210 (1984)
McAdams, D.: Isotone equilibrium in games of incomplete information. Econometrica 71, 1191–1214 (2003)
McAdams, D.: Monotone equilibrium in multi-unit actions. Rev. Econ. Stud. 73, 1039–1056 (2006)
Milgrom, P., Shannon, J.: Monotone comparative statics. Econometrica 62, 157–180 (1994)
Milgrom, P.R., Weber, R.J.: Distributional strategies for games with incomplete information. Math. Oper. Res. 10, 619–632 (1985)
Morris, S.: Purification. In: Durlauf, S., Blume, L. (eds.) The New Palgrave Dictionary of Economics, pp. 779–782. Palgrave Macmillan, New York (2008)
Oechssler, J., Riedel, F.: Evolutionary dynamics on infinite strategy spaces. Econ. Theory 17, 141–162 (2001)
Oechssler, J., Riedel, F.: On the dynamic foundation of evolutionary stability in continuous models. J. Econ. Theory 107, 223–252 (2002)
Radner, R., Rosenthal, R.W.: Private information and pure-strategy equilibria. Math. Oper. Res. 7, 401–409 (1982)
Rashid, S.: Equilibrium points of non-atomic games: asymptotic results. Econ. Lett. 12, 7–10 (1983)
Reny, P.: On the existence of monotone pure-strategy equilibria in Bayesian games. Econometrica 79(2), 499–553 (2011)
Reny, P.J., Zamir, S.: On the existence of pure strategy monotone equilibria in asymmetric first-price auctions. Econometrica 72, 1105–1125 (2004)
Riley, J.G.: Evolutionary equilibrium strategies. J. Theor. Biol. 76, 109–123 (1979)
Ritzberger, K., Weibull, J.: Evolutionary selection in normal-form games. Econometrica 63, 1371–1399 (1995)
Sandholm, W.: Evolution in Bayesian games II: stability of purified equilibria. J. Econ. Theory 136, 641–667 (2007)
Sandholm, W.H.: Population Games and Evolutionary Dynamics. MIT Press, Cambridge (2010)
Sandholm, W.H.: Population games and deterministic evolutionary dynamics. Handb. Game Theory 4, 703–778 (2015)
Schmeidler, D.: Equilibrium points of non-atomic games. J. Stat. Phys. 7, 295–300 (1973)
Vickers, G., Cannings, C.: On the definition of an evolutionary stable strategy. J. Theor. Biol. 129, 349–353 (1987)
Yu, H., Zhang, Z.: Pure strategy equilibria in games with countable actions. J. Math. Econ. 43(2), 192–200 (2007)
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We are particularly indebted to an associate editor and four anonymous referees for their useful suggestions that helped us to improve the paper. All mistakes remain ours.
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Appendices
Appendix 1: Lemma 1 and its proof
A key result for the Proof of Theorem 1 is that any set of weakly best responders is a countable union of sets having measure zero. Lemma 1 below provides such result.
The logic of the Proof of Lemma 1 goes as follows. The joint use of quasisupermodularity in actions (AU1) and strict single crossing in players and actions (AU2) is similar to that in Theorem 4 of Milgrom and Shannon (1994), and it allows to arrange multiple best replies of different players in a linear order. The crucial economic assumption is the strict single crossing property in players and actions, which implies that the sets of weakly best replies of any two distinct players intersect at most at an extreme point and hence are—roughly speaking—rather separated one from the other. The technical assumptions on countability (AA3) and separation (AA4) complete the job, allowing at most a countable number of such sets (see Sect. 4.3 for a discussion on the importance of the countability and separation properties). Therefore, there can exist only a countable number of comparable players that are weakly best responders; for any such player, there can be many (even uncountable) players that are all uncomparable and weakly best responders, but for the comparability assumption (AT2) their measure is null. This leads to conclude that the set of weakly best responders is formed by countably many sets having measure zero, and hence, its measure is zero as well.
Preliminarily, we define \(R_{i,t}(f)\) as the set of best replies to f for \(t \in T_i\), namely \(R_{i,t} (f) = \{a \in A_i :\; u_i(t,a,f_{-t}) \ge u(t,a',f_{-t}) \text { for all } a' \in A_i\}\).
Lemma 1
Let \(\varGamma \) be a game that satisfies AT, AA, and AU. Then, for every \(i \in I\), \(\{t \in T_i:\; ||R_{i,t}(f)|| > 1\}\) is a countable union of sets having measure zero.
Proof
This is the outline of the proof. For a generic \(i \in I\), first we define a function \(g_i\) that maps every \(t \in \{t \in T_i:\; ||R_{i,t}(f)|| > 1\}\) into a pair \((a,a')\) of her best replies, then we define a function \(h_i\), and we use it to assign \((a,a')\) to a base set. We show that function \(h_i\) is injective and that function \(g_i\) is such that any set of players assigned to the same pair of actions has measure zero. Finally, we invoke the fact that there exists only a countable number of base sets to obtain the desired result.
For each \(i \in I\), we consider the partial orders assumed in AA1 (lattice structure) and AT1 (partial ordering) and we take a function \(g_i: \{t \in T_i:\; ||R_{i,t}(f)|| > 1\} \rightarrow A_i^2\) such that \(g_i(t) = (g_{i,0}(t), g_{i,1}(t))\) with \(g_{i,0}(t), g_{i,1}(t) \in R_{i,t}(f)\), \(g_{i,0}(t) <^A_i g_{i,1}(t)\), and \(g_{i,1}(t) \le ^A_i g_{i,0}(t')\) for \(t' >^T_i t\). The following two arguments show that such a function exists for each \(i \in I\). First, \(a \in R_{i,t}(f)\) and \(a' \in R_{i,t}(f)\) imply \(a \vee a' \in R_{i,t}(f)\), so that we can set \(g_{i,0}(t) = a\) and \(g_{i,1}(t) = a \vee a'\), with \(a \vee a'\) existing thanks to AA1 (lattice structure). In fact, \(u_i(t,a,f_{-t}) \ge u_i(t,a \wedge a',f_{-t})\) since \(a \in R_{i,t}(f)\), and hence, \(u_i(t,a \vee a',f_{-t}) \ge u_i(t,a',f_{-t})\) by AU1 (quasisupermodularity in actions), which in turn implies that \(u_i(t,a \vee a',f_{-t}) = u_i(t,a,f_{-t}) = u_i(t,a',f_{-t})\) since \(a \in R_{i,t}(f)\) and \(a' \in R_{i,t}(f)\). Second, \(a \in R_{i,t}(f)\) and \(a' \in R_{i,t'}(f)\) for \(t' >^T_i t\) imply \(a \le ^A_i a'\). This is true since \(u_i(t,a,f_{-t}) \ge u_i(t,a \wedge a',f_{-t})\) due to \(a \in R_{i,t}(f)\), and hence, \(u_i(t,a \vee a',f_{-t}) \ge u_i(t,a',f_{-t})\) by AU1 (quasisupermodularity in actions), and therefore, \(u_i(t',a \vee a',f_{-t}) > u_i(t',a',f_{-t})\) by AU2 (strict single crossing in players and actions), with \(a \wedge a'\) existing thanks to AA1 (lattice structure).
For all \(i \in I\), by AA2 (topology structure), \(A_i\) has a topology and by AA3 (second countability) we can take a countable base \(\mathcal {B}_i\) for such a topology. For each \(i \in I\), we take a function \(h_i: g_i(\{t \in T_i:\; ||R_{i,t}(f)|| > 1\}) \rightarrow \mathcal {B}_i\) such that \(a_1 \in h_i(a_0,a_1)\) and \(a \notin h_i(a_0,a_1)\) for all \(a \le ^A_i a_0\). To see that such a function \(h_i\) exists, note that by AA4 (order separation) for each \((a_0,a_1) \in g_i(\{t \in T_i:\; ||R_{i,t}(f)|| > 1\})\) there exists some open set \(S_{a_1} \subset A_i\) such that \(a_1 \in S\) and \(a \notin S\) for all \(a \le ^A_i a_0\); since \(\mathcal {B}_i\) is a base, there must exist some \(B_{a_1} \in \mathcal {B}_i\) such that \(a_1 \in B_{a_1}\) and \(B_{a_1} \subseteq S_{a_1}\). We set \(h_i(a_0,a_1) = B_{a_1}\).
We check that, for all \(i \in I\), \(g_i\) is such that, for all \((a,a') \in A^2_i\), \(g_i^{-1}(a,a')\) has measure zero. For all \(t,t' \in \{t \in T_i:\; ||R_{i,t}(f)|| > 1\}\), \(t <^T_i t'\), we have that \(g_{i,0}(t) < g_{i,1}(t) \le g_{i,0}(t') < g_{i,1}(t')\) from the definition of function \(g_i\). Therefore, \(t, t' \in g_i^{-1}(a,a')\) implies \(t \not \le ^T_i t'\) and \(t' \not \le ^T_i t\), and AT2 (negligibility of sets of uncomparable players) guarantees that \(\tau _i(g_i^{-1}(a,a'))=0\).
We check that, for all \(i \in I\), \(h_i\) is injective. For all \((a_0,a_1), (a'_0,a'_1) \in g_i(\{t \in T_i:\; ||R_{i,t}(f)|| > 1\})\), \((a_0,a_1) \ne (a'_0,a'_1)\), we know that either \(a_0 < a_1 \le a'_0 < a'_1\) or \(a'_0 < a'_1 \le a_0 < a_1\). Suppose, without loss of generality, that \(a_0 < a_1 \le a'_0 < a'_1\). Then, by the definition of function \(h_i\), we know that \(a_1 \in h_i(a_0,a_1)\), \(a'_1 \in h_i(a'_0,a'_1)\), and \(a_1 \notin h_i(a'_0,a'_1)\) since \(a_1 \le a'_0\). Hence, \(h_i(a_0,a_1) \ne h_i(a'_0,a'_1)\).
Therefore, \(g \circ h\) maps \(\{t \in T_i:\; ||R_{i,t}(f)|| > 1\}\) into \(\mathcal {B}_i\) in such a way that for every \(B \in \mathcal {B}_i\) such that there exists \(t \in T_i\) with \(h(g(t)) = B\), we have that \(\tau _i(\{t \in T_i: h(g(t))=B\})=0\). Since \(\mathcal {B}_i\) is countable, we can conclude that \(\{t \in T_i:\; ||R_{i,t}(f)|| > 1\}\) is the countable union of sets having measure zero. \(\square \)
Appendix 2: Proof of Proposition 2
We start by checking that Theorem 1 can be applied to \(\varGamma ^{IS}\). Clearly, \(\varGamma ^{IS}\) is a special case of \(\varGamma ^{I}\). First, we note that \(\varGamma ^{I}\) is a specific instance of \(\varGamma \). To see this, set i’s type space \(T_i = [0,1]^h\), with associated probability space \((T_i,\mathcal {T}_i,\tau _i)\) where \(\mathcal {T}_i\) is the sigma algebra of all Lebesgue measurable subsets of \(T_i\) and measure \(\tau _i\) is the one induced by \(\phi _i\), implying that \(\tau _i\) is atomless since \(\phi _i\) is bounded. Furthermore, set i’s action space \(A_i = A\). Finally, note that utility \(u^I_i\) is a special case of \(u_i\) where the utility of type t does not depend on the actions chosen by other types of the same player role.
We next check that all hypotheses of Theorem 1 are satisfied.
AU is satisfied by assumption.
We check AT. Since \([0,1]^h\) is a partial order, AT1 is satisfied. Take a set \(\widehat{T} \subseteq [0,1]^h\) which is made of types that are all uncomparable. For any \((t_1, t_2, \ldots , t_{h-1}) \in [0,1]^{h-1}\), there exists at most one \(t_h \in [0,1]\) such that \((t_1, t_2, \ldots , t_{h-1}, t_h) \in \widehat{T}\); otherwise, we would have two elements belonging to \(\widehat{T}\) that are comparable. This shows that \(\widehat{T}\) is contained in the graph of a function from \([0,1]^{h-1}\) to [0, 1], which constitutes an hypersurface in \([0,1]^h\). We know that an hypersurface has Lebesgue measure equal to zero and hence \(\widehat{T}\) as well. Therefore, the measure of \(\widehat{T}\) according to the marginal density function \(\phi _i\) is null, since the integration of \(\phi _i\) over a zero-measure set is zero. So, AT2 is satisfied.
We check AA. If A is a finite lattice, then AA1–AA4 hold trivially. If \(A = [0,1]^k\), then AA1 and AA2 are satisfied by considering, respectively, the standard order and the Euclidean topology on \([0,1]^k\). It is well known that the Euclidean space (and any of its subsets) is second countable (it is enough to consider as base the set of all open balls with rational radii and whose centers have rational coordinates). So AA3 is also satisfied. Finally, consider \(a, a' \in [0,1]^k\) such that \(a'_i \ge a_i\), \(a' \ne a\). Then take an open ball centered at \(a'\) with radius lower than the Euclidean distance between \(a'\) and a; clearly, \(a'\) belongs to the ball, while every \(a'' \in [0.1]^k\) such that \(a''_i \le a_i\) does not belong to the ball. This shows that AA4 is satisfied.
So, we can apply Theorem 1 to conclude that every pure-strategy Nash equilibrium must be essentially strict and monotone in types and actions.
Consider now a symmetric pure-strategy Nash equilibrium where every player chooses strategy \(\alpha \). Consider also any strategy \(\alpha '\), with \(\alpha ' \ne \alpha \). We have already shown, by exploiting Theorem 1, that \(\alpha \) is essentially strict, and so \(u^I(t,\alpha (t),\varvec{\alpha }_{-i}(\mathbf {t}_{-i})) > u^I(t,\alpha ' (t),\varvec{\alpha }_{-i}(\mathbf {t}_{-i}))\) for almost all \(t \in [0,1]^{h}\). Therefore,
which means that \(V(\alpha , \alpha ) > V(\alpha ', \alpha )\). Hence, for \(\epsilon \) small enough, we can conclude that \((1 - \epsilon ) V(\alpha , \alpha ) + \epsilon V(\alpha , \alpha ') > (1 - \epsilon ) V(\alpha ', \alpha ) + \epsilon V(\alpha ', \alpha ')\). We have so established that \(\alpha \) is an ESS.
Finally, to show that an ESS exists, we can rely on Theorem 1 in McAdams (2003) that can be applied since AU2 implies the single crossing condition—which is required by the Theorem. Such theorem, if applied to symmetric games, establishes the existence of a symmetric pure-strategy Nash equilibrium.Footnote 24 By the previous argument, we conclude that the strategy played in such equilibrium must be an ESS.
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Bilancini, E., Boncinelli, L. Strict Nash equilibria in non-atomic games with strict single crossing in players (or types) and actions. Econ Theory Bull 4, 95–109 (2016). https://doi.org/10.1007/s40505-015-0090-8
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DOI: https://doi.org/10.1007/s40505-015-0090-8