Evaluating information in zero-sum games with incomplete information on both sides

In a Bayesian game some players might receive a noisy signal regarding the speciﬁc game actually being played before it starts. We study zero-sum games where each player receives a partial information about his own type and no information about that of the other player


Introduction
In strategic interactions some relevant aspects of the environment might be imperfectly known to players.However, in order to maximize their payoffs, players might look for additional information.The impact of different information on the outcome of a strategic interaction is the subject of this paper.
The set-up we chose to analyze this question is of games with incomplete information.Before the game starts each player obtains a noisy signal about the state of nature.The players' signals are determined by an information structure that specifies how they stochastically depend on the state of nature.Upon receiving the signals the players take actions and receive payoffs that depend on the actions taken and on the state of nature.
In general, evaluating the impact of the information structure on the outcome of the game is not an easy task for a few reasons.First, the interpretation of 'outcome' depends on the solution concept applied.Second, for most solution concepts there are typically multiple outcomes (equilibria).Finally, players might get correlated information, which typically has a significant effect on the outcome.These reasons do not exist when dealing with zero-sum games.In these games, there is one natural solution concept, the value, which induces a unique payoff.
There are two main approaches to analyze the connection between information and payoffs.The first is to compare between two information structures and find which is payoff-wise better than the other.This direction has been widely studied in the literature.Blackwell (1953) initiated this direction and compared between information structures in one-player decision problems.He proved that an information structure always yields a higher value of the problem than another one if and only if it is more informative in a certain sense.Blackwell's result was extended to zero-sum games by Gossner and Mertens (2001).It is well known that this property does not extend to non-zero-sum games and various attempts have been made to understand when it might be extended (see for instance, Hirshleifer (1971), Bassan et al. (2003), Kamien and al. (1990), Neyman (1991), Lehrer and al. (2006)).
The second approach is to study the impact of an information structure on the outcome of the interaction.A typical question in this line is whether the outcome 1 Document de Travail du Centre d'Economie de la Sorbonne -2009.35depends in any particular and discernable way on the information.Since any specific game might have its own idiosyncrasies, an insight into the subject can be obtained only by looking at all possible interactions.
For a given game we define the a value-of-information-function which associates any information structure with the value of the corresponding Bayesian game.We study the properties that are common to all such functions and thereby, the connection between information and outcomes that is not specific to a particular game.Gilboa and Lehrer (1991) treated deterministic information structures and characterized those functions that are value-of-information-functions of one-player decision problems.Blackwell (1953)'s result implies that monotonicity is a necessary condition but it turns out not to be sufficient.Their result has been extended to random information structures by Azrieli and Lehrer (2004).Lehrer and Rosenberg (2006) studied the nature of value-of-information-functions in two cases: (a) one-sided deterministic information: the game depends on a state of nature which is partially known only to player 1 (i.e., player 2 gets no information); and (b) symmetric deterministic information: the game depends on a state of nature which is equally known to both players.In case (a) the more refined the partition the higher the value.Lehrer and Rosenberg (2006) showed that any function defined over partitions which is increasing with respect to refinements is a value-of-informationfunction of some game with incomplete information on one side.
This paper extends the discussions to the case where both players get some partial information on the state of nature.We focus on zero-sum games with lack of information on both sides and independent information.The type k of player 1 and the type l of player 2 are drawn independently of each other.Each player obtains no information about the other player's type and just a partial information about his own.This special case of independent types has been extensively studied in the context of repeated games with incomplete information (see Aumann and Maschler (1996), Mertens and Zamir (1971)).
Formally, the information structure is characterized by a pair of probability transitions (µ, ν) where µ is a probability transition between K and the set of messages of player 1 and ν is a probability transition between L and the set of messages of player 2. Each player is informed only of his message which therefore endows him with some partial information on k for player 1 and l for player 2.Then, a zero-sum game whose payoffs depend on the players' types is played.the information structure, (µ, ν).The function that associates to every information structure the corresponding value is called the value-of-information-function of this game.The goal of this paper is to characterize those functions that are a value-ofinformation-function of some game.
The result concerning games with one-sided information is certainly relevant to the current two-sided information case.In the one-sided and deterministic information case it is proved that a function is a value of information function if and only it is increasing when the information of the informed agent is refined.When the lack of information is on both sides, we show that any value of information function has to be increasing when player 1's information gets refined (and decreasing when the information of player 2 is refined).The notion of refining information has two meanings: monotonicity related to Blackwell's partial order over information structures and concavity over the space of conditional probabilities.That these conditions are necessary is a consequence of known results.Our contribution is to prove that they are sufficient.
The implication of this result is that essentially no further condition beyond monotonicity with respect to information is required to characterize the value-ofinformation functions.This means that the model of Bayesian game with varying information structures can be refuted only by observations that contradict monotonicity.
The center of the proof is the duality between payoffs and information.This means that giving more information to a player amounts to giving him more payoffs in some sense (see also Gossner (2006) (2002)).For any game with incomplete information on one side, he defined a dual game for which the value is the Fenchel conjugate of the value of the initial game.
The paper is organized as follows.We first present the model with the notion of game and of information structure.We define the value of information function.We introduce the notion of standard information structure so that we can state properly the main result that characterizes value of information functions.We define the main notions of concavity, convexity and Blackwell monotonicity, and state the result.We then sketch the proof and state the main structure theorem before proceeding to the proof itself.

Game and information structures
Two players play a zero-sum game with action spaces A and B and whose payoff function g depends on a pair of parameters (k, l), which is randomly chosen from K ×L according to an independent distribution p⊗q.The sets K and L are assumed to be finite and the distributions p and q are assumed to have a full support.
Before the game starts the players obtain a noisy signal that depends on the realized parameter.When (k, l) is realized, player i obtains a signal m i from measurable space (M i , M i ) that is randomly chosen according to the distribution ν i = ν i (h), where h = k, l, depending on i. ν 1 (resp.ν 2 ) is thus a transition probability from K called the information structure of player i and I i denotes the set of all player i's information structures.We will see that the relevant aspect of an information structure is that it induces a probability distribution over conditional probabilities over the state space.Upon receiving the message m 1 player 1 computes the conditional probability that the realized state is k.However, the message m 1 is random and hence, the conditional probability on K given m 1 is random so that I 1 actually induces (ex ante) a distribution over ∆(K).

Strategies and payoffs
Player 1 has action set A and player 2 action set B. The action spaces can contain infinitely many actions and we therefore have to assume that they are endowed with σ-algebras A and B. 1 .Also, for all k, l, the payoff map (a, b) → g(a, b, k, l) ∈ R is 1 We further will make the technical hypothesis that (A, A) and (B, B) are standard Borel.A probability space (A, A) is standard Borel if there is a one to one bi-measurable map between (A, A) and a Borelian subset of R.This hypothesis will ensure the existence of regular conditional probabilities, when required.We could have dispensed with this hypothesis at the price of restricting the definition of a strategy in the next section: a strategy σ would be defined in this context as a measurable map from ([0, 1] × M 1 , B [0,1] ⊗ M 1 ) to (A, A), where B [0,1] is the Borel σ-algebra on [0, 1].Upon receiving the message m 1 , player 1 selects a uniformly distributed random number u ∈ [0, 1] and plays the action σ(u, m 1 ).

4
Document de Travail du Centre d'Economie de la Sorbonne -2009.35assumed to be A⊗B-measurable.A game Γ is thus represented by Γ = A, A, B, B, g .Upon receiving the messages player 1 takes an action a ∈ A and player 2 takes an action b ∈ B. Then player 2 pays player 1 g(a, b, k, l).Note that g depends on both actions and on the parameters k and l.Players are allowed to make mixed moves and a strategy of player 1 is thus a transition probability σ : M 1 → A while a strategy of player 2 is a transition probability τ : M 2 → B. In other words, for all m 1 , σ(m 1 ) is a probability on (A, A) according to which player 1 will choose his action if he receives the message m 1 .Furthermore, for all A ∈ A, the map m 1 → σ(m 1 )[A ] is measurable.A similar condition applies to τ .
A strategy σ (resp.τ ) induces, together with p, ν 1 (resp.q, ν 2 ), a probability distribution over M 1 × A × K (resp.M 2 × B × L), which we denote by Π σ (resp.Π τ ).We also denote π σ and π τ the marginals of Π σ and Π τ on A × K and B × L However, since we deal with general functions g, this expectation could fail to exist for all σ, τ .A strategy σ of player 1 is called admissible if it induces payoffs bounded from below for any strategy of player 2.
Admissible strategies of player 2 are defined in a similar fashion2 .The sets of admissible strategies are denoted Σ a and T a .3

Value of information function
The game with the information structures and I 1 and I 2 is denoted by Γ Our goal is to study the relationship between the value of the game and the information structures.
if there is a game Γ such that for every I 1 , I 2 , the value of the game Γ(I 1 , I 2 ) is The purpose of this paper is to characterize the value-of-information functions.

Standard information structures
The relevant aspect of an information structure is that it induces a probability distribution over conditional probabilities over the state space.Upon receiving the message m 1 player 1 computes the conditional probability that the realized state is k, denoted by [I 1 ](m 1 )(k).Since the message m 1 is random, I 1 actually induces (ex ante) a distribution over4 ∆(K).We refer to this distribution as the law of I 1 and denote it by [I 1 ].Denote by ∆ p (∆(K)) the set of all probability measures ν over ∆(K) whose average value is p (i.e., E ν (p) = p).Note that [I 1 ] belongs to this set.
Different values of m 1 could lead to the same conditional probability [I 1 ](m 1 ).
There is thus apparently more information embedded in m 1 than in [I 1 ](m 1 ) alone.However, this additional information is irrelevant for predicting k since k and m 1 are independent given [I 1 ](m 1 ).Therefore, after computing [I 1 ](m 1 ), player 1 may forget the value of m 1 and play in a game Γ a strategy that just depends on [I 1 ](m 1 ).5 The standard information structure I 1 that is associated to I 1 is defined as follows.The message set is M 1 := ∆(K) and the message is m 1 := [I 1 ](m 1 ) when the message under I 1 was m 1 .Clearly, the posterior probability on K given m 1 is in this case equal to m 1 .It is therefore more convenient to use the letter P for the message sent by I 1 , since it is precisely the posterior.The standard information structure I 2 for player 2 corresponding to I 2 is defined in the same fashion and the message sent by I 2 will be denoted Q.
In a game Γ, player i will play identically with I i an with I i and have the same This implies that the value of information function depends only on the standard information structure, or equivalently on the law induced by the information structure on ∆(K).This fact is captured in the following proposition.
The rest of the paper is devoted to characterizing value of information functions by focusing on functions defined over distribution over posteriors.

The main theorem
Before we state the main theorem, we need a few definitions concerning the notions of concave-convex functions and of Blackwell monotonicity.These turn out to be central in the characterization of value-of-information functions.

Concavity-convexity and Blackwell monotonicity
Definition 3.1 Let µ 1 , µ 2 be two probability measures in ∆ p (∆(K)).We will say that µ 1 is more informative than µ 2 , denoted µ 1 µ 2 , if there exist two random variables Note that 'being more informative than' is a partial order.
Blackwell (Blackwell, 1953) proved that µ 1 is more informative than µ 2 iff in any one-player optimization problem (with the state space being K with probability p) the optimal payoff corresponding to the law of µ 1 is no lower than that corresponding to the law of µ 2 .
to the weak topology on ∆ p (∆(K)) and increasing with respect to .
Note that the definition involves two distinct properties: concavity-convexity and Blackwell monotonicity.Both are known to be related to the fact that information is valuable in zero-sum games.The concavity-convexity property of the value appears in the context of repeated games with incomplete information (see, Aumann and Maschler, 1995).Concavity of a function v states that when µ and µ are distributions over ∆(K), the value the function takes at λµ + (1 − λ)µ , v(λµ + (1 − λ)µ ) is higher than the weighted average of the values it takes at µ and µ , λv(µ) + (1 − λ)v(µ ).We interpret this property assuming that v is the value of a game.Let I 1 be an information structure.Its law, [I 1 ], represents what player 2 knows about player 1's information.Suppose The fact that player 2 knows the information structure means that he knows that the information of player 1 is determined by a lottery that determines whether the law will be µ (with probability λ) or µ (with probability 1 − λ).But player 2 does not know the outcome of this lottery.When player 2 is better informed (knowing the outcome of the lottery), the value of the game, , is smaller (recall, player 2 is the minimizer) than when he is less informed, v([I 1 ]).This is precisely the concavity condition.
Blackwell monotonicity, on the other hand, relates to the information owned by the player about the state space he is directly informed of.Blackwell monotonicity compares the value the function takes at distributions µ 1 and µ 2 that relate to each other in the following way.There exists a random vector (X 1 , X 2 ) such that E[X 1 |X 2 ] = X 2 and X i is µ i distributed, i = 1, 2. This means that the conditional probability over K given by µ 1 is more precise than the one given by µ 2 (it is a martingalization), and the value of the game will therefore be higher.6 None of of concavity-convexity and Blackwell monotonicity properties implies the other.To illustrate the difference between them consider the following example.Let This function is linear in µ and, in particular, concave.Suppose now that µ 2 is a Dirac mass at p and µ 1 ∈ ∆ p (∆(K)) puts no weight at p. Clearly, µ 1 is more informative than µ 2 .However, f (µ 1 ) < 0 = f (µ 2 ).It implies that f is not Blackwell monotonic.

Main theorem
The main result of this paper which will be proved in the next sections is: ) for a concave-convex, semi-continuous and Blackwellmonotonic function v.
Blackwell proved that monotonicity is a necessary condition for a function to be the value of a one-player optimization problem.Since the value of a game is also the optimal value of an optimization problem of player i, assuming implicitly that player −i plays a best reply, Blackwell monotonicity is a necessary condition here also.Lehrer and Rosenberg (2006) proved that this is also the case in games with incomplete information on one side with deterministic information.
As in Lehrer and Rosenberg (2006), Theorem 3.3 proves that only the concavityconvexity property and Blackwell monotonicity are common to all value-of-information functions.It implies that when one observes the results of interactions under various information functions, the only properties of the observed results that one may expect to obtain are concavity-convexity and Blackwell monotonicity, which are necessary.Thus, the main implication of Theorem 3.3 is that only violations of concavityconvexity property or Blackwell monotonicity can refute the hypothesis that the interaction can be modeled as a game played by Bayesian players.For an elaboration of this point, see Lehrer and Rosenberg (2006).

The deterministic case
In the special case of deterministic information, more can be said.It turn out that in this case, the concave-convex property and Blackwell-monotonicity coincide.
When µ 1 and ν 2 are restricted to be deterministic functions, it is equivalent for player 1 to know m 1 or to know the subset of K for which the message would be m 1 .A deterministic information structure can be therefore modeled as a pair of partitions P, Q respectively on K and L with the interpretation that if the true state is k, l player 1 is informed of the cell of P that contains k and player 2 is informed of the cell of Q that contains l.We denote the set of partitions on K and L respectively by K and L. Note that there are only finitely many such partitions.A value of information function is now a real function defined over the set of partitions.In this case Blackwell monotonicity is translated to the following.Recall that a partition P is said to refine a partition P if any atom of P is a union of atoms of P.
A function V from K × L to R is increasing (resp.decreasing) in P (resp.Q) if for any P, P in K, such that P refines P , and any Theorem 3.4 Let v be a function on pairs of partitions respectively on K and L. It is a value of information function iff it is Blackwell monotonic.Moreover it can then be obtained as the value of a finite game (i.e., A and B are finite).
This theorem differs from Theorem 3.3 in two respects.First, the games are finite.
This stronger result can be derived from Theorem 3.3, taking into account that there are only finitely many partitions.Second, the condition of concavity-convexity of the function is dropped here.This is so because any function of partitions that is Blackwell monotonic can be extended to a function over laws of posterior probabilities that is concave-convex, semi continuous and Blackwell monotonic (this result is proved in section 10).Thus, Theorem 3.4 derives from Theorem 3.3 which extends Lehrer and Rosenberg (2006) that applies to games with incomplete information with deterministic information.

A structure theorem
The first main step of the proof is to prove a structure theorem that characterizes concave-convex and Blackwell monotonic functions.
This theorem extends the well-known characterization that states that a convex function is the maximum of the linear functions that are below it.In our setup a linear function on µ (a distribution over ∆(K)) is the integral of some function on ∆(K) with respect to µ.Therefore, a concave function f satisfies that for any law µ 0 on ∆(K), f (µ 0 ) is the infimum of ψ(p)dµ 0 (p) over continuous functions ψ (defined on ∆(K))that satisfy ∆(K), E µ [ψ(P )] ≥ f (µ) for every law µ on ∆(K).
The second part of the theorem characterizes functions that have the concaveconvex property and are also Blacwell monotonic.It gives a clearer sense of the difference between both notions.In order for a function to be in addition Blackwell monotonic it needs to be the infimum of ψ(p)dµ(p) over continuous functions ψ such that (i) the integral is greater than f (µ), and (ii) ψ is convex.The intuition behind this result is that ψ(p) represents the payoff that can be achieved by some strategy of player 2 when the conditional probability on K is p and player 1 plays optimally.The integral ψ(p)dµ(p) represents the expected payoff and the minimum is taken with respect to available strategies of player 2. Blackwell monotonicity implies that a martingalization of the conditional probabilities gives more information to player 1 and enables him to increase the payoff so that ψ will be convex.
In order to state formally the result, we need a few notations.
Ψ 0 and Ψ 1 denote respectively the set of continuous and continuously differentiable functions from ∆(K) to R. The set of convex functions in Ψ i will be denoted Ψ i,vex , i = 0, 1.For a given function ψ ∈ Ψ 0 , and a law µ in ∆ p (∆(K)), we denote by ψ(µ) the expectation of ψ with respect to µ (i.e., ψ(p)dµ(p)).
5 The conditions of Theorem 3.3 are necessary In this section we show that any value-of-information function is concave-convex and Blackwell monotonic.
) is a value of information function, then v must be concave-convex semi-continuous and Blackwell monotonic.
Indeed, one can compute the expected payoff given a pair of strategies (σ, τ ), denoted by g(σ, τ ).One expression of this payoff is where σ(P ) denotes the strategy followed given the conditional probability is P .Note that since σ is the family of all possible σ(P ) for all P , one has This function is the expectation of a convex function in P so that by Jensen's inequality it is Blackwell increasing in [I 1 ] and so is v Moreover the above expression proves that sup σ g(σ, τ ) is linear and continuous in ) is concave and usc.
6 Sketch of the proof of Theorem 3.3 Our goal is now to prove the reciprocal of the above theorem.For an arbitrary concave-convex, Blackwell monotonic and semi-continuous function v, we construct a game Γ such that 6.1 Games with incomplete information on one side

A first construction: one-sided information game
We first sketch the proof of the main result in the special case where L is a singleton, which is the case of games with incomplete information on one side.In this case the value functions depend only on their first argument.Given a concave, semi-continuous and Blackwell monotonic function v, we construct a two stage game whose value is The first intuition is to construct a two stage game.At the first stage player 2 picks a function ψ ∈ Ψ 1,vex v , and his choice is observed by player 1.Then from stage 2 on, the game will proceed in a way that is continuation value will be ψ(µ).The structure theorem then implies that the value of such a game is indeed given by v.
12 Document de Travail du Centre d'Economie de la Sorbonne -2009.35In order for the value of the continuation game to be ψ(µ), we use the convexity of ψ and allow player 1 to choose a payoff a k for each state k, with the constraint that the overall expected payoff is dominated by ψ.This amount to choosing a linear function of the conditional probability P on K that is dominated by ψ.
At the first stage Player 2 selects ψ ∈ Ψ 1,vex v .Player 1 is then informed of ψ.The message given by player 1's standard information structure (which is defined in section 2.4) is the conditional probability P on k given his message.He then selects an affine functional a(P, ψ), p + α(P, ψ) dominated by ψ.The payoff of the game is a(P, ψ), P + α(P, ψ)7 .
The normal form of this game has value v, because of the constraint on the choices of α and a and of the definition of Ψ 1,vex v .

Another parametrization of the same game
Observe that when informed of ψ, P , player 1 may select an affine functional tangent to ψ at some point p(P, ψ), for otherwise his strategy would be dominated.Since ψ is differentiable, the corresponding affine functional of p is ψ(p(P, ψ))+ ∇ψ(p(P, ψ)); p− p(P, ψ) .One might thus think of the previous game as the following two stage game.
Player 2 selects ψ ∈ Ψ 1,vex v .Player 1 is informed of ψ and P , and then he selects an action p ∈ ∆(K).The payoff is given by ψ(p(P, ψ)) + ∇ψ(p(P, ψ)); P − p(P, ψ) , Put it differently, if the chosen action are ψ, p and the state is k, the payoff is This game formulation has the advantage that player 1's action space is independent of player 2's move.

A simultaneous-move game
In order to extend the previous construction to games with incomplete information on both sides, player 2 should be given a way to condition his move on his information.
However, letting player 2 play first will enable player 1 to extract information from the chosen act of player 2. For this reason we need to construct a simultaneous move game with the same value.
The simultaneous move game is described as follows.Once informed of P , Player 1 selects a point p(P ) ∈ ∆(K).Simultaneously, Player 2 picks ψ ∈ Ψ 1,vex v and the payoff g(k, p, ψ) is given by ( 2).Note that this game has the same value like the previous one, because in the previous game an optimal strategy for player 1 was to ignore ψ and to choose p(ψ, P ) = P .

The extension to games with incomplete information on both sides: a first intuition
We turn now to games with incomplete information on both sides.We will find a game Γ whose value is v( From the section on games with incomplete information above, we know that for a fixed [I 2 ], one can construct a game Γ [I 2 ] with value v.However, the strategy space of player 2 in Γ One way to unify the strategy spaces in Γ [I 2 ] is the following.If ψ ∈ Ψ 1,vex , then the function ψ + α belongs to Ψ 1,vex v(•,ν) whenever the constant α is large enough.More precisely, we require that ∀µ ∈ ∆ p (∆(K)) : ψ(µ) + α ≥ v(µ, ν).This turns out to be equivalent to The idea is to let player 2 choose any function ψ ∈ Ψ 1,vex and to use ψ + w in the previous game, leading thus to a payoff of: Note that the function w is the Fenchel transform of the value v.This Fenchel transform can be viewed as the value of a dual game.This might help interpreting it as the value of information function of a game.

Reminder on dual games
The notion of the dual game was first introduced by De Meyer.Suppose that v is a value of information function, and G is the game whose value is V .Furthermore, suppose that ψ is a function on ∆(K) that we will call a cost.In the dual game there are two stages.First, player 1 can buy an information structure µ at the cost of ψ(µ) and this choice is publicly observed.Then both players take part in the game G and get the corresponding payoff.Therefore the overall payoff in the dual game is the sum of the payoff in the original game that has been played and the cost of the information structure µ that has been chosen.The idea is that on the one hand it is always better for player 1 to get more information but on the other hand since this information is costly, depending on the cost, he might choose to buy a less informative structure.
It turns out that the value of such a game is the Fenchel conjugate of v defined by Moreover w is convex and decreasing for the Blackwell order in [I 2 ].

The construction of the game
We now turn to sketching the final proof in the case of games with lack of information on both sides.
For i = 1, 0, Φ i,cav f will denote the set of concave functions φ ∈ Φ i f .
chooses ψ ∈ Ψ 1,vex , and the payoff is This property enables us to separate ψ and [I 2 ] and therefore to avoid the definition of a game that would depend on ψ and yield a two stage game.
From the above we know that φ([I 2 ]) can be interpreted as the value of a game with incomplete information on one side, where player 2 is the informed player.
In words, each player selects two items.The first is a fictitious conditional probability to which he plays a best response as if this probability was the true one.The second is a payoff function for his opponent so that playing a best response with respect to this function leads to the desired value-of-function function by the structure theorem.

Intuition of a construction through dual games
Another approach is to try to construct the game using known results on games with incomplete information on one side.Indeed, the result on games with incomplete information on one side proves that for each ψ, w(ψ, .) is a value of information function of a game with incomplete information on one side where player 2 is the informed player (call G ψ the corresponding game) and that ψ is a value of information function of a game with incomplete information on one side where player 1 is the informed player (call G the corresponding game).The idea is to define a game in which first player 2 chooses ψ and this choice is observed, then games G and G are played independently for this choice of ψ and the total payoff is the sum of payoffs.
The problem with such a construction is that in the true game player 2 can use his information to choose ψ and in order for the value not to be affected by this possibility one has to prove that it is optimal for player 2 not to use his information in the first stage.This can be done at least in the deterministic case.As we did in the case of games with incomplete information on one side we may also avoid this problem by designing a simultaneous move game.
For i = 1, 0, Ψ i,vex f will denote the set of convex functions ψ ∈ Ψ i f .Recall Theorem 4.1.
3. In this case, we also have ∀µ : Proof: We prove the first claim.Let f be a concave weakly-usc function on ∆ p (∆(K)).Since ∆ p (∆(K)) is weakly compact, f is bounded from above so that Ψ 0 f = ∅.The definition of Ψ 0 f implies that ∀µ, f (µ) ≤ inf ψ∈Ψ 0 f ψ(µ) As for the converse inequality, for a given µ ∈ ∆ p (∆(K)), let t be such that Note that α > 0. Otherwise, one of the α i should be strictly positive ((α 1 , . . ., α n , α) = 0) and the left hand side of the last inequality evaluated at µ = µ would then be strictly positive.This is a contradiction.
We now prove claim 2. Let f be in addition Blackwell increasing.Let ψ be in Ψ 0 f and let φ := vex(ψ) be the convexifiaction of ψ.That is, φ is the largest lsc convex function dominated by ψ.Since for all x in ∆(K) there is a probability distribution µ x on ∆(K) such that E µx [p] = x and that φ(x) = E µx [ψ(p)], for any random variable X with probability distribution µ in ∆ p (∆(K)), we may consider a random vector Y Claim 3 follows from the fact that a convex continuous function ψ on ∆(K) can be uniformly approximated by a function ψ 2 ∈ Ψ 1,vex , up to an arbitrary > 0. Indeed, ψ is the supremum of the set G of affine functionals g dominated by ψ on ∆(K).For G , and we conclude that ψ 1 (p) := max{g(p) + /2 : g ∈ G } is a convex function that satisfies ψ ≤ ψ 1 ≤ ψ + /2.The function ψ 1 , as a maximum of finitely many affine functional is the restriction to ∆(K) of a Lipschitz concave function defined on R K (also denoted ψ 1 ).Let us next define, for an arbitrary ρ > 0, ψ 2 (x) := E[ψ 1 (x + ρZ)], where Z is a random vector whose components are independent standard normal random variables.Then ψ 2 is continuously differentiable due the smoothing property of the convolution operator.ψ 2 is also convex, since if λ ∈ [0, 1], x, x ∈ R K : From Jensen's inequality, we get ψ 1 (x) ≤ ψ 2 (x) ≤ ψ 1 (x) + ρκE[ Z ], where κ is the Lispchitz constant of ψ 1 .For ρ small enough, we get thus ψ ≤ ψ 2 ≤ ψ + .
8 The duality between v and w We here prove the results stated in Section 6.3: the duality relation between v and w as well as the dual structure theorem.
w is thus a function on U := Ψ 0 × ∆ q (∆(L)).It is convenient to introduce the following sets.Φ 0 and Φ 1 refer to the sets of functions φ on ∆(L) that are respectively continuous and continuously differentiable.For i = 1, 0, Φ i,cav will denote the set of concave functions in Φ i .
The expected payoff conditionally to (P, Q) is then k,l P k Q l g(k, l, a P , b Q ) = ψ Q (p P ) + ∇ψ Q (p P ), P − pP − ψQ (µ P ) + φ P (q Q ) + ∇φ P (q Q ), Q − qQ Proposition 2.2 If V is a value of information function, then there exists a map Document de Travail du Centre d'Economie de laSorbonne -2009.35

7
Document de Travail du Centre d'Economie de la Sorbonne -2009.35