All Games Have Equilibria Note: The authors gratefully acknowledge with warm gratitude Sam Alexander, Justin Bledin, Michael Grossberg, Joe Lawlor, Rich McLean, Larry Moss, Rohit Parikh, Kris Patel, Andrew Powell, Kevin Reffett, Phil Reny, Eddie Schlee, Joel Sobel, Jack Stecher, and Metin Uyanik. Khan should also acknowledge Jeremy Goodman’s workshop Topics in Epistemology in the spring of 2024. Preliminary versions of this paper were presented by Stinchcombe at seminars at UT Austin and the University of California at Riverside; he thanks participants for their active engagement. This version is to be presented at the Fourth International Workshop on Game Theory and Economic Applications to be held at the University of São Paulo, from July 26 to August 2, 2026.

Much of the prior work with finitely additive strategies is of doubtful merit for reasons both obvious and hidden. In the obvious cases, central claims are categorically false. In the other cases, some of the work has implicitly confined attention to games isometrically isomorphic to compact games with jointly continuous payoffs, nullifying any presumption to greater generality achieved by way of finitely additive strategies. Other parts of this work have adopted theories of integration that systematically skew payoffs against deviations to deliver “equilibria” with payoffs that are not close to being feasible. Still other parts of this work redefine “equilibrium” in ways that risk compromising a game’s strategic integrity or fail to determine equilibrium utility levels. The remaining work makes use of correlating devices to determine payoffs that may skew payoffs against deviations and thereby deliver equilibria unrelated to the games’ strategic structures.

Karlin (1950, Theorem 10, p. 152-3) claims that for any bounded measurable kernel $k(x,y)$ on $X\times Y=[0,1]\times[0,1]$, there exist finitely additive probabilities $\mu_{1}$ and $\mu_{2}$ constituting a Nash equilibrium for the zero-sum game with strategy sets $X=Y=[0,1]$ and utilities $(k(x,y),-k(x,y))$. The claim is false. This is demonstrated in §4.5.2 which shows, inter alia, how far off one goes by trying to use iterated integrals to define payoffs. Yanovskaya (1970, p. 153) astutely traces the source of Karlin’s error to the implicit use of Fubini’s theorem. As this fails for finitely additive measures, the order of integration in Karlin’s optimization problems matters for the payoffs. She argues that, since the order of integration in the players’ optimization problems may determine payoffs, one should interpret the result as yielding existence for the sequential games with one or the other player moving first and the other responding.

In pursuing equilibrium existence within the finitely additive framework, the literature has had to contend with the failure of Fubini’s theorem. The conditions imposed to avoid that failure have, necessarily, returned the analysis to the Glicksberg (1952) setting of compact metric spaces and jointly continuous payoffs. Some background for this statement is in order.

Pták (1964, Theorem 2.2, p. 567) establishes criteria for the joint continuity of a separately continuous function on a product space. Simons (1968, Theorem 7) and Sinclair (1974, Theorem 4.4) deploy this result to characterize functions for which there is order-independent equality of iterated integrals for all finitely additive probabilities. Young (1971, Theorem 5, p. 199) and Thomsen (1978, 4, pp. 421-2) apply these criteria to recover the restricted class of zero-sum games for which Karlin’s result holds. Fenstad (1967) arrives at the same equilibrium existence result by a more direct measure-theoretic route.

The condition isolated by Fenstad (1967) was independently rediscovered by Marinacci (1997) and, in a publication-delayed contribution, by Harris et al. (2005). The latter gave equivalent algebraic, measure theoretic, functional analysis and finite approximability characterizations of the condition. The most revealing result in Harris et al. (2005) shows that, after identification of strategically equivalent strategies, the class of games satisfying this condition is isometrically isomorphic to the class of compact games with jointly continuous payoffs. This is how this branch in the study of finitely additive strategies for games returns precisely to where it began, with the Glicksberg (1952) class of compact and continuous games.

An alternative response to the failure of Fubini’s theorem is to redefine the equilibrium concept itself. Flesch et al. (2017) pursues this for zero-sum games, Vasquez (2017, §1), for finite player games, and Flesch et al. (2021) for games with arbitrary player sets. Each evaluates equilibrium utilities by the upper integral while evaluating deviation utilities to the lower integral.¹¹ 1 For these integrals, see e.g. Royden (1988, §4.2) or Billingsley (2012, Problems 15.1-7). This systematically severs the connection between equilibrium utility levels and the strategic structure of the game: in zero-sum games, both players may receive strictly positive payoffs in equilibrium, directly violating the antagonistic structure that zero-sum payoff functions encode; in the standard Bertrand model of price competition for a homogeneous good, both firms may receive monopoly profits in “equilibrium,” vitiating the competitive pressure the model is designed to capture. In each case, the equilibrium utility levels so obtained fail to be feasible payoffs for the game.

Another change in the definition of equilibrium is in Milchtaich (2023). He defines a “best-response equilibrium” in finitely additive strategies for a subclass of games. To be a Milchtaich best-response equilibrium, a vector of mixed strategies must mesh with the utility functions in two ways. First, for each player $i$ and each of their actions, $a_{i}\in A_{i}$, the function $a_{-i}\mapsto u_{i}(a_{i},a_{-i})$ be integrable with respect to the product of the other players’ finitely additive probabilities. That is, it requires that $v_{i}(a_{i})=\int u_{i}(a_{i},a_{-i})\,d\mu_{-i}(a_{-i})$ be well-defined.²² 2 For games with $2$ players, this is no restriction. For games with $3$ or more players, this requires equality of some of the iterated integrals giving payoffs and this defines the subclass of games to which one can apply the solution concept. Second, defining $\overline{v}_{i}=\sup_{a_{i}\in A_{i}}v_{i}(a_{i})$, each player’s mixed strategy must put zero mass on the set of $a_{i}$ for which $v_{i}(a_{i})<\overline{v}_{i}-\epsilon$ for each $\epsilon>0$. While this looks very similar to definitions that work well for countably additive strategies that are jointly measurable with respect to the product $\sigma$-field, as he notes (p. 5), $\overline{v}_{i}$ “cannot generally be interpreted as player $i$’s equilibrium payoff” (emphasis in the original). For example, it allows the ‘equilibria’ for the standard Bertrand price competition model, and the counterexample to Karlin’s claims in §4.5.2 shows that the $\overline{v}_{i}$ need not sum to $0$ in zero-sum games.

The remaining works, Yanovskaya (1970), its special case in Schervish and Seidenfeld (1996), and Stinchcombe (2005, §5), all deliver finitely additive equilibria by using correlating devices to determine payoffs at discontinuity points. This is best understood as a species of the endogenous sharing rule equilibria of Simon and Zame (1990), and we will discuss them in more detail in that context. In sum, two essential problems attend this approach. First, the resulting equilibria may place unit mass on strictly dominated strategies. Second, much as with the upper and lower integral constructions just described, the randomization governing payoffs may differ systematically between equilibrium play and deviations. That said, this approach does deliver payoffs summing to $0$ in zero-sum games and it delivers well-defined equilibrium utility levels.

A related but distinct strand of work takes correlation itself as the object of analysis. Hart and Schmeidler (1989) establish existence of correlated equilibria for finite games and extend the result to games with infinitely many players, finite action sets, and bounded measurable payoffs; they further remark, without proof, that an analogous result holds for finitely additive correlated equilibria in the compact Hausdorff setting (p. 24). In their formulation, the equilibrium object is a joint distribution on the product of the action spaces: correlation across players is constitutive of the solution concept, not a device for determining payoffs. The set of correlated equilibria always contains the set of Nash equilibria and in general is strictly larger; Hart and Schmeidler themselves observe that the finitely additive correlated equilibria so obtained can be “quite unreasonable” (p. 22).

To be sure, the equilibrium concept adopted here also departs from the classical definition: mixed strategies are finitely additive, and equilibrium is defined as immunity to pure strategy deviations. The restriction is principled: finitely additive deviations must themselves be defined by specifying the limit process by which they are reached, and the finitely approximable equilibria of Theorem C do exactly this. The three requirements on equilibrium analysis — existence, stability, and operationalizability — are not compromised.

The present paper delivers what the prior literature could not. The equilibrium correspondence is nonempty valued and upper hemicontinuous, and the equilibria it delivers are immune to any pure strategy deviation. Equilibrium is a product extension of individual mixed strategies, and every deviation is evaluated against the same product measure — the contrast with Hart and Schmeidler, who define equilibrium on joint distributions and accept correlation, is exact: finite additivity is the instrument in both cases, but what it operates on determines the strategic content of the resulting equilibria. No restrictions beyond boundedness are imposed on the utility functions, foreclosing any inadvertent return to the compact and continuous setting of Glicksberg (1952). The strategic structures built into the games survive intact.

Without compactness, the existence of countably additive equilibria can no longer be taken for granted. The literature has addressed this challenge in three ways: (i) by imposing boundary conditions on utility functions that push best responses toward compact sets, an approach directly analogous to the stability conditions for Markov chains with continuum state spaces; (ii) by finding conditions sufficient for the existence of approximate equilibria and asking whether their limits are themselves equilibria; and (iii) by compactifying the strategy spaces so that those limits are explicitly represented as points in a well-defined compact space. The three approaches overlap considerably, and each is subsumed within our framework.

Meyn and Tweedie (2009), in their monumental study of the stability of Markov chains with locally compact state spaces, identify a family of sufficient conditions for stochastic stability. Each is an implementation of the same governing idea: that transition probabilities push the system back toward compact sets whenever the state drifts too far from them. Replacing transition probabilities with best responses or approximate best responses carries this idea directly into game theory.

For game and general equilibrium models, perhaps the most direct statement of this principle is “Of course, in order to guarantee the existence of equilibria on a non-compact set, some kind of ‘boundary’ assumptions (i.e., assumptions on utility functions outside of some compact set) are absolutely necessary.” Tian (1992a, p. 380-1, Theorem 2(iii)) gives the boundary requirement its most direct and explicit form, and the closely related Tian (1992b, Theorem 2) generalize much of the earlier work on relaxing compactness by giving this principle a precise form. Tian (2015, Theorem 3.2) carries the program further still, establishing equilibrium existence by positing a compact subset on which the conditions sufficient — and necessary — for countably additive equilibrium existence are satisfied.

When boundary conditions cannot be guaranteed, exact countably additive equilibria may fail to exist, and the literature has turned instead to approximate ones. Wald (1945, §3) shows that approximate countably additive equilibria exist for zero-sum games in which one player has a finite set of actions and the other a countably infinite one. Yanovskaya (1974, §2) surveys the broader literature on zero-sum games. With $X$ and $Y$ denoting the set of countably additive mixed strategies for the two players, she catalogues sufficient conditions for

This condition is weaker than the existence of an exact countably additive equilibrium,

but strong enough to guarantee the existence of countably additive $\epsilon$-equilibria for all $\epsilon>0$. When $(\dagger)$ holds but $(\ddagger)$ does not, there is a gap between having a value and having an equilibrium. From Corollary B.1, we know that this gap does not exist in our approach.

Tijs (1981) extends the approximate equilibrium existence results for zero-sum games to general finite player games, and in doing so draws the connection between the two approaches into sharp relief. His central observation is that finite approximations to the strategy sets that nearly deliver best responses to all strategies are themselves an instance of the requirement that best responses push players toward compact sets — here, finite ones. Equicontinuity of the utility slices, or uniform continuity of the utility functions on uniformly bounded action spaces, guarantees the existence of such approximations, and hence the existence of countably additive $\epsilon$-equilibria for all $\epsilon>0$. What these approximations do not supply is the limiting strategies themselves. This lack is something that compactification fills by enlarging the strategy spaces so that all limits of approximating strategies are explicitly represented, giving the limits of approximate equilibrium strategies a precise mathematical home.

Young (1937) pioneered this approach in the calculus of variations, embedding measurable functions from $[0,T]$ to a set of distributions on a compact set of actions $A$ as distributions on their graphs in the compact space of probabilities on $[0,T]\times A$ that have the uniform distribution as the marginal on $[0,T]$. Milgrom and Weber (1985) used the same compactification in game theory to study games where players choose their mixed strategies as a measurable function of the information contained in their signal. Lacker (2015) and Lacker (2021) extend the reach of this compactification further still, deploying Young measures in the analysis of mean field games and the convergence of finite player games to their mean field limits.

The compactification used in this paper embeds each player’s set of mixed strategies as a dense subset of the set of finitely additive probabilities defined on the class of all subsets of actions, a space sufficiently rich to represent all limits of approximate equilibria along nets of finite approximations. This represents all vectors of mutual best responses identified in the boundary condition approach and all limits of vectors of approximate mutual best responses identified in the approximation approach.

The finitely additive probabilities can be identified with supnorm continuous linear functionals on the set of all bounded functions. Restricted to vector subspaces of functions containing the utility slices, these linear functionals are integrals against probabilities — countably additive ones when the equilibria of the boundary condition approach are recovered, and finitely additive ones when the limits of the approximate equilibria of the approximation and compactification approaches are represented. The present framework thus delivers what the boundary condition, approximate equilibria, and compactification approaches each deliver, and more.

With compactness and countable additivity assumed, the question becomes what class of discontinuities of the utility functions still encompass games of economic interest while still being restrictive enough to guarantee countably additive equilibrium existence. Yanovskaya (1974) provides a masterful survey of the early work on zero-sum games, and Radzik (1996) provides a more detailed historical focus on games of timing. While many of the techniques and themes — upper semicontinuity, quasi-concavity, and diagonal transfer continuity — recurred in the study of general games, Radzik and Ravindran (1989) showed that conditions sufficient for equilibrium existence in zero-sum games need not remain sufficient in the general setting.

The early literature pursued equilibrium existence by studying limits of equilibria for games played on sequences of finite approximations to the action spaces. By Nash’s theorem, equilibria exist for each finite approximation; by compactness, any sequence of such equilibria has an accumulation point. For an accumulation point to be an equilibrium of the limit game, two conditions must be met: the limits of the finite equilibrium utilities must be utilities in the limit game; and the payoffs to deviations must not jump upward. The later literature abandoned finite approximations in favor of direct existence arguments, identifying conditions on the discontinuities that permit a fixed-point argument to go through directly. We examine these two strands directly below (§2.3.1 and §2.3.2).

There is, however, a fundamental tension running through both. The countably additive mixed strategies on compact metric spaces are defined to converge when their integrals against all continuous functions converge. But this notion of convergence is applied to discontinuous utility functions, and all of the difficulties in this literature arise because these two are not generally compatible. The equilibrium utilities along a convergent sequence of equilibrium strategies need not converge; and even when they do, the payoffs to deviations may jump upward in the limit, destroying the immunity to pure strategy deviations built into the definition of equilibria. The present paper resolves this tension entirely by moving to finitely additive probabilities and defining convergence by the convergence of integrals against all bounded functions.

The literature just surveyed restricts attention to utility functions with special discontinuities. By contrast, the sharing rule approach of Simon and Zame (1990) confronts wayward discontinuities directly — recasting them through a limit and convexification process that redefines the utility function at the discontinuity points.

The construction proceeds in three steps. The first step closes the graph of the utility, yielding a payoff correspondence that is multi-valued precisely at the discontinuity points. The second step convexifies the range of this closed-value correspondence, accommodating limits of arbitrarily correlated randomization near the discontinuities. And the third step shows that there exists a measurable selection from the resulting convex-valued correspondence with the property that the game with those payoffs has a countably additive equilibrium, called a sharing rule, or selection equilibrium.

The construction purchases equilibrium existence at a price — but it offers something in return. It offers a finitely additive interpretation, connecting it to the correlating device equilibria of Yanovskaya (1970), Schervish and Seidenfeld (1996), and Stinchcombe (2005) briefly discussed in §2.1. The price is threefold: the tie-breaking rules it imposes may not respect independence in the random choices across the players; they may ignore strictly dominant strategies, yielding “equilibria” in strictly dominated strategies; and they may evaluate equilibrium strategies and deviations using different correlating devices — the same dependence that severs the connection to any recognizable notion of Nash equilibrium flagged in §2.1.

The arbitrariness of the tie-breaking rules cuts deepest in games where they are not a technical artifact but an essential feature of the game’s specification — auctions foremost among them. To change them is to analyze a different game; and if the equilibria depend on that change, they have nothing to say about behavior in the situation being modeled. This is not always the case, but it requires separate and often subtle arguments. For example, for a large class of auction models, Jackson and Swinkels (2005) show that the choice of sharing rule values at the discontinuities does not matter.

Selection rules may ignore strictly dominated strategies, yielding equilibria that play them. Stinchcombe (2005, Cor. 3.3.1, p. 347) shows that this can be remedied by replacing sequences of finite approximations with exhaustive nets of finite approximations — precisely the approach adopted here (see Definition 3.4 below). More subtle is the problem of differing correlating devices. As Stinchcombe (2005, Example 2.4 and §2.5.2, pp. 339-40) shows, the limit correlation embodied in the convexification step can differ at deviations from what it is at putative equilibrium strategies — the same asymmetry between equilibrium utility evaluation and deviation utility evaluation that, as in parts of the literature on finitely additive probabilities has done, severs the connection to Nash equilibrium.

The present paper sidesteps these costs entirely. By working with finitely additive mixtures, we have no need to change the utility functions — neither at the discontinuities nor anywhere else. The utility functions are taken as given, the tie-breaking rules are not imposed, and the evaluation of equilibrium strategies and deviations is governed by the same integral throughout.

The finitely additive interpretation flagged above is made precise in Stinchcombe (2005, §5). Drawing on the representation theory of Yosida and Hewitt (1952, §4) and recapitulated in the appendix, arbitrary strategy spaces are embedded as a dense subset of a compact space whose points are identified with the zero-one (Z1) probabilities. The utility function in the original game is then extended by the same selection equilibrium logic, and these selections capture precisely the correlation lost in the passage to the limit when countably additive strategies are used. The result is the existence of finitely additive equilibria, but ones that inherits the weaknesses of the sharing rule construction from which they proceed.

Yanovskaya (1970) and Schervish and Seidenfeld (1996) pursue the same embedding for zero-sum games, but with further restrictions: the latter works in a rather difficult-to-identify subset of the zero-sum games, while the former treats all zero-sum games. They both use mid-point rules to determine limit utilities at discontinuities, and they both assign their single choice of payoff to all discontinuity points independent of the nearby payoffs. The result is a class of equilibria related to sharing rule equilibria defined by a particular tie-breaking convention rather than by the strategic logic of the game itself.

It is here that the present paper’s results find their sharpest expression. The finitely approximable equilibria of Theorem C admit a sharing rule interpretation, but without play of strictly dominated strategies, without arbitrary tie-breaking rules, and without the asymmetry between the evaluation of equilibrium strategies and deviations that the sharing rule construction allows.

The following defines the class of games studied here.

There are no assumptions on the cardinality of the set of players. There are no topological or measure theoretic assumptions on the sets of actions. There are no assumptions on the utility functions except boundedness: the uniformity of the bound across agents is without loss of generality; and some bound is necessary to preclude phenomena similar to the St. Petersburg paradox.

The starting point is the set of total probabilities. When we turn to games, the set $X$ in the following will be $A=\times_{i\in I}A_{i}$.

By induction, if $\{B_{n}:n=1,\ldots,N\}$ is a finite collection of disjoint sets, a total probability must satisfy $p(\cup_{n=1}^{N}B_{n})=\sum_{n=1}^{N}p(B_{n})$.

Nash equilibria require independent randomization by the players. The existence of the following extensions is guaranteed by Lemma 3.1.

We use the following game-theoretic notation, for $a\in A$, ${i\in I}$, and $b_{i}\in A_{i}$, the point $a\backslash b_{i}\in A$ is defined by $\operatornamewithlimits{proj}_{j}(a\backslash b_{i})=a_{j}$ for $j\neq i$ and $\operatornamewithlimits{proj}_{i}(a\backslash b_{i})=b_{i}$. And we extend this notation to mixtures over $A$, for $\mu$ a probability on $A$, ${i\in I}$ and $b_{i}\in A_{i}$, $\mu\backslash b_{i}$ is the image measure of $\mu$ under the mapping $a\mapsto a\backslash b_{i}$ from $A$ to $A$.

Finitely additive probabilities are often understood as having “various peculiar properties” (e.g. Yosida and Hewitt (1952, p. 56)). One might worry that the use of such probabilities would deliver an equilibrium theory that is not recognizable. The next section analyzes several examples with a view to understanding how to work with finitely additive equilibria while the following result offers some theoretical reassurance.

To focus on the dependence on the utility function, we denote the game $\Gamma=(A_{i},u_{i})_{{i\in I}}$ as $\Gamma(u)$ where $u:A\rightarrow[-B,+B]^{I}$, and for each game $\Gamma(u)$, $Eq(u)\subset\Delta$ denotes the set of equilibria and $u\mapsto Eq(u)$ is the equilibrium correspondence. We define the convergence of utility functions by $u^{\alpha}\rightarrow u$ if (and only if) for all ${i\in I}$, the supnorm distance, $\|u_{i}^{\alpha}(\cdot)-u_{i}(\cdot)\|$, converges to $0$. The proof of the following assertion is a minor variant on the textbook arguments for finite games.

There are many games for which there are countably additive $\epsilon$-equilibria for all positive $\epsilon$, but for which there does not exist a countably additive equilibrium. By contrast, the logic of the proof of Theorem B immediately delivers the following.

There are many examples of games for which there are approximate countably additive equilibria but no countably additive equilibrium.¹⁰¹⁰ 10 Bertrand price competition for a homogenous good but with different marginal costs is such a game. Glicksberg (1950) shows that zero-sum games on compact metric spaces having upper semi-continuous payoffs for player $1$, hence lower semi-continuous payoffs for player $2$, can belong to this class. Other examples include Laraki et al. (2005), which gives Markov perfect $\epsilon$-equilibria for every $\epsilon>0$ for general timing games, Barelli et al. (2014, Theorem 2.9, p. 280), which gives $\epsilon$-equilibria for every $\epsilon>0$ in games modeling competitions for a majority.

We analyze the examples in the next section by replacing each player’s set of actions by an exhaustive net of finite approximations and studying the limits of equilibria after iterated deletion of weakly dominated strategies in the net of approximating finite games. Theorem C below shows that this process always delivers a non-empty, closed set of equilibria with a well-behaved equilibrium correspondence.

Our use of finitely supported probabilities in the following is neither usual nor without loss of generality, but it is appropriate for our exhaustive finite nets approach to finitely additive equilibria.

Given weak dominance, iterated weak dominance is defined as usual.

For a game with bounded payoffs $\Gamma(u)=(A_{i},u_{i})_{{i\in I}}$, let $Eq^{Fin}(\Gamma(u))$ denote the set of limits of equilibria for the finite games $(F_{i}^{\alpha},u_{i})_{{i\in I}}$ along any exhaustive net $\alpha\mapsto\times_{i\in I}F_{i}^{\alpha}$ of finite approximations to $\times_{i\in I}A_{i}$, and let $Eq^{iwu}(\Gamma(u))\subset Eq^{Fin}(\Gamma(u))$ denote the set of limits of iteratively weakly undominated equilibria for those finite games. Note that along any exhaustive net of finite approximations, any iteratively weakly dominated strategy is eventually excluded from every finite approximation to the game along the net.

The existence proof directly parallels the proof of Theorem A and the rest of the arguments directly parallel the arguments for Theorem B. The next section mostly analyzes the set $Eq^{iwu}(\Gamma(u))$, which is generally a proper subset of the finitely approximable equilibria, which is, in turn, generally a proper subset of the finitely additive equilibria.

The asymmetric location game is the simplest example in which the payoff discontinuity at a single point precludes a countably additive equilibrium while the finitely additive framework delivers one immediately. Consumers are distributed uniformly on $[0,1]$. Licensing restrictions confine player $1$ to locations in $A_{1}=[0,0.8]$ and player $2$ to locations in $A_{2}=[0.8,1]$. Given choices $x<y$, each consumer patronizes the nearest location — the consumer at the midpoint $\frac{1}{2}x+\frac{1}{2}y$ is indifferent, and their choice has no effect on payoffs. The discontinuity arises at $x=y=0.8$: the consumers then view the two locations as perfect substitutes, with $\frac{1}{2}$ patronizing each player.

Stinchcombe (2005, Example 2.2, p. 337) makes good on the warning just issued: he presents an example that delivers an endogenous sharing rule “equilibrium” that puts mass $1$ on a strictly dominated strategy, and is therefore not a Nash equilibrium. Stinchcombe (2005, Theorem 3.3, p. 344) shows that the remedy is precisely the one adopted here: exhaustive nets of finite approximations, rather than sequences of finite sets that become dense in the metric topology — a technique whose roots go back to Simon and Stinchcombe (1995). Bich and Laraki (2017, Dfn. 2.16, p. 85) strengthen the definition of a pure strategy endogenous sharing rule equilibrium in a fashion that rules out some of the other problems with the sharing rule construction — but still falls short: it fails to rule out equilibria that play strictly dominated strategies.

The following adaptation of the Stinchcombe (2005) example illuminates precisely why. The game $\Gamma=(A_{i},u_{i})_{i=1,2}$ is described as follows: the action sets are $A_{1}=A_{2}=[0,1]$; player $2$’s utility function is $u_{2}(a_{1},a_{2})=a_{2}$ if $a_{2}>0$, and $u_{2}(a_{1},0)=2$ if $a_{2}=0$ so that every $a_{2}>0$ is strictly dominated by $a_{2}=0$; player $1$’s utility function has two parts, for $a_{1}>0$, $u_{1}(a_{1},a_{2})=a_{1}$, and for $a_{1}=0$,

This game is dominance solvable and the unique equilibrium is $(a_{1}^{*},a_{2}^{*})=(0,0)$ with equilibrium utility levels $u^{*}=(2,2)$.

Specialized to this game, Bich and Laraki (2017, Dfn. 2.16 and 2.17, p. 85–86) define a vector of actions $(b_{1}^{\dagger},b_{2}^{\dagger})\in A_{1}\times A_{2}$ to be a pure sharing rule equilibrium if $(b_{1}^{\dagger},b_{2}^{\dagger})$ is a pure strategy Nash equilibrium for an auxiliary game $\widetilde{\Gamma}=(A_{i},g_{i})_{i=1,2}$ where for all $(b_{1},b_{2})\in A_{1}\times A_{2}$, the auxiliary utility function $g(b_{1},b_{2})\in{\mathbb{R}}^{2}$ belongs to the closure of the graph of the utility function $u$ — unlike Simon and Zame (1990), without convexifying the set of limit payoffs at the discontinuities. For the game described above, the unique continuous selection from the closure of the graph of the utility functions is $g_{1}(a_{1},a_{2})=a_{1}$ and $g_{2}(a_{1},a_{2})=a_{2}$, and the unique Nash equilibrium of the auxiliary game is $(b_{1}^{\dagger},b_{2}^{\dagger})=(1,1)$ with utility levels $u^{\dagger}=(1,1)$ — a strictly dominated strategy profile delivering utilities strictly below the unique equilibrium $u^{*}=(2,2)$.

The inadequacy of metric density is on full display in the following observations. Let $n\mapsto(F_{1}^{n},F_{2}^{n})$ be a sequence of finite sets for which $d(F_{1}^{n}\times F_{2}^{n},[0,1]\times[0,1])\rightarrow 0$.

1. 1.

   If $F_{2}^{n}$ fails to contain the dominant strategy $a_{2}=0$, then the unique limit of approximate equilibrium play is the sharing rule “equilibrium” $(b_{1}^{\dagger},b_{2}^{\dagger})=(1,1)$, which plays a strictly dominated strategy.

2. 2.

   If $n\mapsto F_{2}^{n}$ contains $0$ for all large $n$ but $0\notin F_{1}^{n}$, then the unique limit of approximate equilibria is the endogenous sharing rule “equilibrium” $(c_{1}^{\dagger},c_{2}^{\dagger})=(1,0)$ with payoffs $(1,2)$, which involves a strictly dominated response for player $1$.

3. 3.

   If $\alpha\mapsto(F_{1}^{\alpha},F_{2}^{\alpha})$ is any exhaustive net of finite approximations to $[0,1]\times[0,1]$, then the unique limit of approximate equilibria is the unique Nash equilibrium for the game, $(a_{1}^{*},a_{2}^{*})=(0,0)$ with equilibrium payoffs $(2,2)$.

The contrast between Obs 1–2 and Obs 3 identifies the essential problem precisely. The sharing rule construction is adapted to arguments that apply to all approximating sequences of finite subsets that become dense in the metric topology — but metric density has no bearing on whether dominant strategies or strict best responses are included when the utility functions are not continuous. Exhaustive nets, by contrast, guarantee that every action is eventually included in every finite approximation, and it is this guarantee — not metric density — that delivers the correct equilibrium.

Sion and Wolfe (1957) give an asymmetric, two battlefield, Colonel Blotto game where both players have one unit of force to allocate between the battlefields. The player allocating the larger/smaller force to a battlefield wins/loses, and equal force allocations lead to ties. The asymmetry is that player $2$ starts with an advantage of $0.5$ immobile units of force already present in the second battlefield. The payoffs are additive, with $+1$ for each battlefield won, $-1$ for each one lost, and $0$ for ties.

Let $x$ and $(1-x)$ denote player $1$’s allocations to the first and second battle fields respectively, and let $y$ and $(1-y)$ denote the corresponding allocations for player $2$. The payoffs are:

To make the payoffs stay in the interval $[-1,+1]$, Sion and Wolfe (1957) add $+1$ to player $1$’s payoffs and $-1$ to player $2$’s payoffs so that utilities are given by:

and $u_{2}(x,y)=-u_{1}(x,y)$. Diagramatically, we can represent $u_{1}(\cdot,\cdot)$ as in Figure 1.

Sion and Wolfe (1957) show that this game has no countably additive $\epsilon$-equilibria for a range of strictly positive $\epsilon$. We give two finitely additive equilibria that represent the limits of “reasonable” finitely supported equilibria for this game, providing a contrast with both the Sion and Wolfe result and the literature on discontinuous games that has assiduously avoided games of this sort.

The differences between the two finitely additive equilibria depend on details of the hyperfinite action sets, equivalently on the details of the net of finite approximations. Player $2$’s advantage of $0.5$ in the second battlefield means that fine details of the approximation around $(0.5,0.5)$ can matter. Let $H_{1}$ and $H_{2}$ denote the two players’ exhaustive hyperfinite action sets, and throughout, let $h_{i}^{\prime}$ denote player $i$’s largest strategy strictly less than $0.5$ in $H_{i}$.

We analyze two cases, the fully symmetric one, and one of the asymmetric ones. In the fully symmetric case, we suppose that $H_{1}=H_{2}$, equivalently, that the nets of finite approximations satisfy $F_{1}^{\alpha}=F_{2}^{\alpha}$. In the asymmetric case, we suppose that $h_{1}^{\prime}<h_{2}^{\prime}$, equivalently, that $h_{2}^{\alpha}$, the largest elements of $F_{2}^{\alpha}$ below $0.5$, is larger than the corresponding $h_{1}^{\alpha}$ in $F_{1}^{\alpha}$. It is of particular note that we find equilibria with different values. This is consistent with the point of view that continuum games are not fully specified until one has specified what large finite sets the continuum is supposed to represent.

The Sion and Wolfe game makes plain why the representation of “just under” matters: the equilibrium depends not merely on whether a player concentrates mass below $r$, but on which player can approach $r$ more closely. Three representations of this idea have appeared in the analysis — exhaustive nets, exhaustive hyperfinite sets, and finitely additive limits. This subsection shows that all three are equivalent. Mutatis mutandis, the same analysis covers numbers just over an $r\in[0,1)$.