Introduction

1The first formulation of correlated equilibrium by Aumann [1974] models randomization devices explicitly and allows for players to have different subjective probability assessments about these randomization devices. Today, one usually uses the canonical formulation of correlated equilibrium introduced later by Aumann [1987], which disposes of both subjectivity and the specification of an underlying randomization device. [1]

2Aumann [1974] also proved a number of results in which the subjectivity of probability assessments plays little role. This is possible because there are enough events on which all probability assessments agree. At the heart of Aumann’s argument is his Lemma 4.1 [2] to the effect that if (T, Σ) is a measurable space and µ₁,…, µ_n a finite family of atomless probability measures, then there exists for each α ∈ [0, 1] a measurable set E ∈ Σ such that µ_i(E) =…= µ_n(E) = α. The proof is based on Lyapunov’s convexity theorem. If each µ_i represents the subjective belief of player i, the lemma shows that one can construct an objective event with any prescribed probability and, consequently, a finite measurable partition with any prescribed distribution on which all beliefs agree. The same result was also proven in the context of fair division by Dubins and Spanier [1961]. Later, Border, Ghirardato and Segal [2008] used the result for finite partitions to show that a σ-algebra Σ′ ⊆ Σ exists on which all µ_i agree and on which the µ_i have range [0, 1]. Their goal was to find agreeable probabilities for randomized social choice. As is shown below, the same construction allows for an even stronger result: The µ_i do not just have range [0, 1]; they are actually atomless. The strengthened statement is Proposition 1 below. It is this stronger statement that can be used to generalize various arguments by Aumann [1974].

Existence of Objective Lotteries

3We take all measures to be countably additive. Let (\(\Omega, \mathcal{B}, \mu\)) be a probability space. A µ-atom is a set \(A \in \mathcal{B}\) such that μ(A) > 0 and such that for all \(B \in \mathcal{B}\) satisfying B ⊆ A either μ(B) = μ(A) or μ(B) = 0. We say that µ is atomless if there is no µ-atom. It is well known that the range of an atomless probability measure is the whole unit interval; see Theorem 10.52 of Aliprantis and Border [2006]. If \(\mathcal{B}^{\prime}\) is a sub-σ-algebra of \(\mathcal{B}\), we say that µ is atomless on \(\mathcal{B}^{\prime}\) if the restriction of µ to \(\mathcal{B}\) is atomless.

4The σ-algebra constructed in the following result is constructed in exactly the same way as by Border, Ghirardato and Segal [2008].

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Proposition 1. Let µ₁,…, µ_n be atomless probability measures on a common measurable space (Ω, ∑). Then there exists a sub-σ-algebra ∑ʹ ⊆ ∑ on which all measures agree and are atomless.

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Proof. Using Lemma 4.1 or the result of Dubins and Spanier, we can partition Ω into two measurable sets on which all the measures agree and take on a value of 1/2. We can further partition these sets to obtain four measurable sets on which all the measures agree and take on a value of 1/4. We continue this way and take finite unions of all sets thus obtained. This gives us a countable sub-algebra \(\mathcal{A}\) of Σ on which all measures agree. So there is a unique extension to the σ-algebra \(\sigma(\mathcal{A})=\Sigma^{\prime}\) generated by \(\mathcal{A}\) by the Carathéodory extension theorem. Consequently, all measures agree on \(\sigma(\mathcal{A})\).
Let µ be the common restriction of the µ_i to \(\sigma(\mathcal{A})\). We show that µ is atomless. Let \(B \in \sigma(\mathcal{A})\) with μ(B) > 0. Choose m such that 1/2^m < μ(B) and partition Ω into 2^m disjoint measurable sets B_l, l = 1,…, 2^m, of equal µ-measure 1/2^m. By construction, this can be done. Now,
\(\mu(B)=\sum_{l=1}^{2^m} \mu\left(B \cap B_l\right).\)
So there exists l such that μ(B ∩ B_l) > 0. Since μ(B ∩ B_l) ≤ μ(B_l) = 1/2^m < μ(B), there exists a measurable subset of B with smaller but positive measure.

7One can show that for a given atomless probability space, any Borel probability measure on a Polish (separable and completely metrizable) space is the distribution of a random variable on the given probability space. [3] Proposition 1 allows us, therefore, to generate arbitrary distributions on such spaces that are agreed on by everyone.

Applications to Subjective Correlated Equilibria

8This section’s notation, terminology, and setting follow Aumann [1974] closely. However, Aumann calls actions pure strategies.

9A game consists of

1. A finite set N = {1, …, n} of players;
2. For each i ∈ N, a nonempty, compact, and metrizable space A_i of actions of i;
3. A compact, and metrizable space X of outcomes endowed with the Borel σ-algebra;
4. A continuous outcome function g from \(A=\prod_{i \in \in N} A_i\) onto X.

10All these spaces were additionally assumed to be finite by Aumann [1974]. To this description of the direct strategic environment, we add the following:

5. A nonempty set Ω of states of the world endowed with a σ-algebra \(\mathcal{B}\) containing the events;
6. For each i ∈ N, a sub-σ-algebra \(\mathcal{I}_i\) of \(\mathcal{B}\) representing the information of i;
7. For each i ∈ N, a preference order \(\succeq_i\) on the set of measurable functions from Ω to X, the lotteries.

11Throughout, we make the following assumptions:

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Assumption 1. For each i ∈ N, there exists a continuous utility function u_i : X → ℝ and a unique probability measure p_i on (\(\Omega, \mathcal{B}\)), such that for all lotteries x and y, we have \(\mathbf{x} \succeq_i \mathbf{y}\) if and only if
\(\int_{\Omega} u_i \circ \mathbf{x} \mathrm{d} p_i \geq \int_{\Omega} u_i \circ \mathbf{y} \mathrm{d} p_i.\)
As usual, N_-i stands for N \ {i}. An event \(E \in \mathcal{I}_i\) is i-secret if it is p_j-independent of all events in \(\sigma\left(\bigcup_{j \in N_{-i}} \mathcal{I}_j\right)\) for all j ∈ N_-i. So an i-secret event is an event that i is informed about, but that other players cannot learn anything about even if they would pool all their information. The family of i-secret events is denoted by \(\mathcal{S}_i\). In general, \(\mathcal{S}_i\) need not be a σ-algebra.

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Assumption 2. For each i ∈ N, there exists a σ-algebra \(\mathcal{R}_i \subseteq \mathcal{S}_i\) on which p_j is atomless for all j ∈ N.
An event E is objective if p₁(E) = p₂(E) = …= p_n(E). We let O be the family of all objective events. In general, O need not be a σ-algebra. An event is subjective if it is not objective. A function from Ω to a measurable space is objective if it is O-measurable, that is, if the preimages of measurable sets are objective. A strategy of i ∈ N is a function s_i : Ω → A_i that is I_i-measurable. The strategy s_i is mixed if it is also S_i-measurable. This represents the idea that a player’s mixed strategy, the way the term is usually used in game theory, chooses independently of what other players are doing. Usually, this is done in terms of product distributions; here, it is done in terms of σ-algebras because all randomized action choices are given in terms of random variables on an underlying space.
We identify a profile of strategies with a measurable function s : Ω → A. For each strategy profile s and i ∈ N, we let
\(H_i(\mathbf{s})=\int_{\Omega} u_i \circ g \circ \mathbf{s} \mathrm{d} p_i.\)
We let H(s) = (H₁(s),…, H_n(s)) and call each point of the form H(s) a feasible payoff. We say the strategy profile s is a subjective correlated equilibrium or simply an equilibrium if for every i ∈ N and all strategies t_i of i, we have
\(H_i(\mathbf{s}) \geq H_i\left(\left(\mathbf{s}_1, \ldots, \mathbf{s}_{i-1}, \mathbf{t}_i, \mathbf{s}_{i+1}, \ldots, \mathbf{s}_n\right)\right).\)
A point of the form H(s) with s being an equilibrium is an equilibrium payoff.
A Nash equilibrium payoff is simply the vector of the usual expected payoffs in a game with player space N, and player i having strategy space S_i and payoff-function u_i ∘ g. We first prove that the usual Nash equilibria embed as equilibria in the present framework, and thus, by the Fan-Glicksberg fixed-point theorem, equilibria do exist. The following is essentially Lemma 7.3, slightly restated.

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Lemma 1. Fix a player i. For every j ≠ i, let \(\mathcal{W}_j \subseteq \mathcal{S}_j\) be a σ-algebra. Then the σ-algebras \(\mathcal{W}_1, \ldots, \mathcal{W}_{i-1}, \mathcal{I}_i, \mathcal{W}_{i+1}, \ldots, \mathcal{W}_n\) are µ_i-independent.

15We can now generalize Proposition 4.3. and Proposition 5.1, respectively, by Aumann [1974] to a continuum of actions.

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Proposition 2. The set of Nash equilibrium payoffs coincides with the set of equilibrium payoffs of equilibria in which all strategies are objective and mixed.

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Proof. First, let s be an equilibrium in objective mixed strategies and i ∈ N. Then s_i is a best response to s_–i and, a fortiori, a best response within the class of objective mixed strategies. Since all p_j are atomless on \(\mathcal{R}_i\), there exists by Proposition 1 an atomless sub-σ-algebra \(\mathcal{W}_i\) of \(\mathcal{R}_i\) on which all p_j agree and are atomless. Therefore, any best response of i in the classical setting is the distribution of an objective mixed strategy measurable with respect to \(\mathcal{W}_i\). By the last lemma, only the distribution is relevant for it to be a best response for i; independent randomizations correspond to product measures as distributions. So H(s) is a Nash equilibrium payoff.
Similarly, we can again apply Proposition 1 to find for each i ∈ N a sub- σ-algebra \(\mathcal{W}_i\) of \(\mathcal{R}_i\) on which each p_j agrees and is atomless. If we have any Nash equilibrium, (μ_i)_i∈N in the classical game, we can find a \(\mathcal{W}_i\)-measurable strategy s_i : Ω → A_i with p_i ∘ s^–1_i = μ_i. The last lemma again shows that these strategies form an equilibrium that, clearly, induces the same expected payoff.

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Proposition 3. Let n = 2 and assume p₁(B) = 0 if and only if p₂(B) = 0 for all \(B \in \mathcal{B}\). Then for any equilibrium s in mixed strategies, there is an equilibrium t in objective mixed strategies such that H(s) = H(t).

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Proof. Let (s₁, s₂) be an equilibrium in mixed strategies. Using Proposition 1 with \(\mathcal{R}_i\), p₁ and p₂, we can find an objective mixed strategy t₁, such that p₂ ∘ s^–1₁ = p₂ ∘ t^–1₁. Similarly, we can find an objective mixed strategy t₂ such that p₁ ∘ s^–1₂ = p₁ ∘ t^–1₂. We show that (t₁, t₂) is an equilibrium and H(s) = H(t). Since s₂ is p₁-independent of \(\mathcal{I}_1\), the expected utility of 1 depends only on the distributions of the strategies of the two players under p₁. Let β be the set of maximizers of ∫u_i ∘ g(⋅, x₂) dp₁ ∘ s^–1₂. Then β is compact and a strategy of player i is a best response to s₂ if and only if its values lie p₁-almost surely in β. Consequently, p₁{ω : s₁(ω) ∈ β} = 1. Since both players agree on all certain events,
\(p_2\left\{\omega: \mathbf{s}_1(\omega) \in \beta\right\}=1=p_2\left\{\omega: \mathbf{t}_1(\omega) \in \beta\right\}=p_1\left\{\omega: \mathbf{t}_1(\omega) \in \beta\right\}.\)
Since s₂ and t₂ have the same distribution for 1, t₁ is also a best response to t₂. Similarly, one can show that t₂ is a best response to t₁ for player 2. Therefore, (t₁, t₂) is an equilibrium.

Limitations

20We can also discuss which arguments do not generalize. These are the arguments related to public randomization. An event E is public if it is in \(P=\bigcap_{i \in N} \mathcal{I}\). A public roulette \(\mathcal{R}\) is a sub-σ-algebra of \(\mathcal{P}\) on which all p_i agree and are atomless. As a direct consequence of Proposition 1, we get the following characterization of the existence of a public roulette.

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Proposition 4. A public roulette exists if and only if p_i is atomless on \(\mathcal{P}\) for all i ∈ N.

22However, public roulettes are less powerful in our context. Aumann shows that the existence of a public roulette implies that for every event B, one can find a public event C with any prescribed probability that is µ_i-independent of B for every i ∈ N; Lemma 4.4. This guarantees the convexity of the set of equilibrium payoffs and the set of feasible payoffs when action spaces are finite; Aumann’s Proposition 4.5. However, the proof relies on every strategy profile being measurable with respect to a finite partition. This then guarantees that one can find public events independent of all strategies. The argument does not generalize to a continuum of pure strategies: Let Ω = [0, 1]³ be the unit cube with \(\mathcal{B}\) the Borel σ-algebra, p₁ = p₂ be the uniform distribution on the unit cube, let \(\mathcal{I}_1\) be the σ-algebra generated by the first two coordinate-functions, and \(\mathcal{I}_1\) be the σ-algebra generated by the second and third coordinate function. Then the set of public events is exactly the set of events in the σ-algebra generated by the second coordinate function. Let A₁ = A₂ = [0, 1], g the identity, and u₁ = u₂ = u be given by u₁(a₁, a₂) = –(a₁ – a₂)². There is an equilibrium in which both players simply play the second coordinate. Since the second coordinate function generates all public events, no public event is independent of the equilibrium strategies. Of course, this example only shows that the particular proof does not generalize.