Comparison of Information Structures for Zero-Sum Games in Standard - - PowerPoint PPT Presentation

Comparison of Information Structures for Zero-Sum Games in Standard Borel Spaces

Ian Hogeboom-Burr and Serdar Yüksel

Queen’s University, Department of Mathematics and Statistics

ISIT 2020

1 / 27

Introduction

Characterizing the value of information is a problem studied in many disciplines involving decision-making under uncertainty. An information structure is said to be better than another if it guarantees the decision-maker (DM) will not perform worse in any valid decision-problem under the former than under the latter. In single-player decision problems, Blackwell’s seminal results [Blackwell ’53] provide necessary and sufficient conditions for comparing two information structures. In general games, such an ordering is generally not possible due to perturbations in equilibria caused by changes in information. There exist games where more information can hurt the decision maker who receives it. Zero-sum games were first studied in this context in [Gossner-Mertens ’01], and necessary and sufficient conditions were found by P˛ eski in [P˛ eski ’08] for games with finite spaces.

2 / 27

Introduction

Characterizing the value of information is a problem studied in many disciplines involving decision-making under uncertainty. An information structure is said to be better than another if it guarantees the decision-maker (DM) will not perform worse in any valid decision-problem under the former than under the latter. In single-player decision problems, Blackwell’s seminal results [Blackwell ’53] provide necessary and sufficient conditions for comparing two information structures. In general games, such an ordering is generally not possible due to perturbations in equilibria caused by changes in information. There exist games where more information can hurt the decision maker who receives it. Zero-sum games were first studied in this context in [Gossner-Mertens ’01], and necessary and sufficient conditions were found by P˛ eski in [P˛ eski ’08] for games with finite spaces.

2 / 27

Introduction

Characterizing the value of information is a problem studied in many disciplines involving decision-making under uncertainty. An information structure is said to be better than another if it guarantees the decision-maker (DM) will not perform worse in any valid decision-problem under the former than under the latter. In single-player decision problems, Blackwell’s seminal results [Blackwell ’53] provide necessary and sufficient conditions for comparing two information structures. In general games, such an ordering is generally not possible due to perturbations in equilibria caused by changes in information. There exist games where more information can hurt the decision maker who receives it. Zero-sum games were first studied in this context in [Gossner-Mertens ’01], and necessary and sufficient conditions were found by P˛ eski in [P˛ eski ’08] for games with finite spaces.

2 / 27

Introduction

Characterizing the value of information is a problem studied in many disciplines involving decision-making under uncertainty. An information structure is said to be better than another if it guarantees the decision-maker (DM) will not perform worse in any valid decision-problem under the former than under the latter. In single-player decision problems, Blackwell’s seminal results [Blackwell ’53] provide necessary and sufficient conditions for comparing two information structures. In general games, such an ordering is generally not possible due to perturbations in equilibria caused by changes in information. There exist games where more information can hurt the decision maker who receives it. Zero-sum games were first studied in this context in [Gossner-Mertens ’01], and necessary and sufficient conditions were found by P˛ eski in [P˛ eski ’08] for games with finite spaces.

2 / 27

Outline

1

Problem Setup

2

Supporting Results

3

Comparison of Information Structures in Zero-Sum Games

3 / 27

A Motivating Example

Consider a card drawn at random from a deck, where its colour can be either red or black, each with probability 1/2. Player 1 first declares his guess of the colour, and then, after hearing what Player 1 guessed, Player 2 submits her guess for the colour. If both players guess the same colour, the payout is $2 each, whereas if one player guesses correctly, that player receives a payout of $6 and the other player receives $0. In the case where both players are uninformed about the colour of the card, the expected payout is $3 each, as Player 1’s optimal strategy is arbitrary, and Player 2’s optimal strategy is to guess the opposite colour of what Player 1 guessed. In the case where both players are informed of the colour of the card prior to declaring their guess, the equilibrium for the game occurs when both players guess the true colour of the card. In this case, the expected payout becomes $2 for each player. This example, adapted from [Bassan-Gossner-Scarsini-Zamir ’01], shows how additional information can negatively affect performance in a general game.

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A Motivating Example

Consider a card drawn at random from a deck, where its colour can be either red or black, each with probability 1/2. Player 1 first declares his guess of the colour, and then, after hearing what Player 1 guessed, Player 2 submits her guess for the colour. If both players guess the same colour, the payout is $2 each, whereas if one player guesses correctly, that player receives a payout of $6 and the other player receives $0. In the case where both players are uninformed about the colour of the card, the expected payout is $3 each, as Player 1’s optimal strategy is arbitrary, and Player 2’s optimal strategy is to guess the opposite colour of what Player 1 guessed. In the case where both players are informed of the colour of the card prior to declaring their guess, the equilibrium for the game occurs when both players guess the true colour of the card. In this case, the expected payout becomes $2 for each player. This example, adapted from [Bassan-Gossner-Scarsini-Zamir ’01], shows how additional information can negatively affect performance in a general game.

4 / 27

A Motivating Example

Consider a card drawn at random from a deck, where its colour can be either red or black, each with probability 1/2. Player 1 first declares his guess of the colour, and then, after hearing what Player 1 guessed, Player 2 submits her guess for the colour. If both players guess the same colour, the payout is $2 each, whereas if one player guesses correctly, that player receives a payout of $6 and the other player receives $0. In the case where both players are uninformed about the colour of the card, the expected payout is $3 each, as Player 1’s optimal strategy is arbitrary, and Player 2’s optimal strategy is to guess the opposite colour of what Player 1 guessed. In the case where both players are informed of the colour of the card prior to declaring their guess, the equilibrium for the game occurs when both players guess the true colour of the card. In this case, the expected payout becomes $2 for each player. This example, adapted from [Bassan-Gossner-Scarsini-Zamir ’01], shows how additional information can negatively affect performance in a general game.

4 / 27

A Motivating Example

Consider a card drawn at random from a deck, where its colour can be either red or black, each with probability 1/2. Player 1 first declares his guess of the colour, and then, after hearing what Player 1 guessed, Player 2 submits her guess for the colour. If both players guess the same colour, the payout is $2 each, whereas if one player guesses correctly, that player receives a payout of $6 and the other player receives $0. In the case where both players are uninformed about the colour of the card, the expected payout is $3 each, as Player 1’s optimal strategy is arbitrary, and Player 2’s optimal strategy is to guess the opposite colour of what Player 1 guessed. In the case where both players are informed of the colour of the card prior to declaring their guess, the equilibrium for the game occurs when both players guess the true colour of the card. In this case, the expected payout becomes $2 for each player. This example, adapted from [Bassan-Gossner-Scarsini-Zamir ’01], shows how additional information can negatively affect performance in a general game.

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Single-Player Setup

Let x ∼ P be an X-valued random variable, where X is a standard Borel space (a Borel subset of a complete separable metric space). We call x the state of nature. Let Y be the standard Borel measurement space for the player. The player makes a measurement y from a measurement channel Q, where Q is a stochastic kernel from X to Y. (Equivalently, we view y = g(x, ν) for some independent measurement noise ν). The objective for the player is the minimization of the expected cost for some measurable cost function c(x, u) : X × U, where U is the DM’s standard Borel action space. This minimization occurs over all measurable policies γ ∈ Γ := {γ : Y → U}. We write the expected cost as: J(P, Q, γ) = EQ,γ

P

[c(x, γ(y)] Given fixed P, X, and Y, a single-player decision problem is a pair (c, U) of a cost function and an action space.

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Single-Player Setup

Let x ∼ P be an X-valued random variable, where X is a standard Borel space (a Borel subset of a complete separable metric space). We call x the state of nature. Let Y be the standard Borel measurement space for the player. The player makes a measurement y from a measurement channel Q, where Q is a stochastic kernel from X to Y. (Equivalently, we view y = g(x, ν) for some independent measurement noise ν). The objective for the player is the minimization of the expected cost for some measurable cost function c(x, u) : X × U, where U is the DM’s standard Borel action space. This minimization occurs over all measurable policies γ ∈ Γ := {γ : Y → U}. We write the expected cost as: J(P, Q, γ) = EQ,γ

P

[c(x, γ(y)] Given fixed P, X, and Y, a single-player decision problem is a pair (c, U) of a cost function and an action space.

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Single-Player Setup

Let x ∼ P be an X-valued random variable, where X is a standard Borel space (a Borel subset of a complete separable metric space). We call x the state of nature. Let Y be the standard Borel measurement space for the player. The player makes a measurement y from a measurement channel Q, where Q is a stochastic kernel from X to Y. (Equivalently, we view y = g(x, ν) for some independent measurement noise ν). The objective for the player is the minimization of the expected cost for some measurable cost function c(x, u) : X × U, where U is the DM’s standard Borel action space. This minimization occurs over all measurable policies γ ∈ Γ := {γ : Y → U}. We write the expected cost as: J(P, Q, γ) = EQ,γ

P

[c(x, γ(y)] Given fixed P, X, and Y, a single-player decision problem is a pair (c, U) of a cost function and an action space.

5 / 27

Single-Player Setup

Let x ∼ P be an X-valued random variable, where X is a standard Borel space (a Borel subset of a complete separable metric space). We call x the state of nature. Let Y be the standard Borel measurement space for the player. The player makes a measurement y from a measurement channel Q, where Q is a stochastic kernel from X to Y. (Equivalently, we view y = g(x, ν) for some independent measurement noise ν). The objective for the player is the minimization of the expected cost for some measurable cost function c(x, u) : X × U, where U is the DM’s standard Borel action space. This minimization occurs over all measurable policies γ ∈ Γ := {γ : Y → U}. We write the expected cost as: J(P, Q, γ) = EQ,γ

P

[c(x, γ(y)] Given fixed P, X, and Y, a single-player decision problem is a pair (c, U) of a cost function and an action space.

5 / 27

Single-Player Setup

The question of comparing two information structures is the following: when can one compare two measurement channels Q1, Q2 such that: inf

γ∈Γ J(P, Q1, γ) ≤ inf γ∈Γ J(P, Q2, γ),

for a large class of single-player decision problems?

Definition 1

An information structure µ is more informative than another information structure Φ if inf

γ∈Γ EΦ,γ P

[c(x, u)] ≥ inf

γ∈Γ Eµ,γ P [c(x, u)]

for some class of single player decision problems.

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Garbling of Channels

The following definition will be essential to many of the results presented.

Definition 2

An information structure induced by some channel Q2 is garbled (or stochastically degraded) with respect to another one, Q1, if there exists a channel Q′ on Y × Y such that Q2(B|x) =

• Y

Q′(B|y)Q1(dy|x), B ∈ B(Y), P a.s. x ∈ X. Visually, if Φ is a garbling of µ:

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Blackwell’s Comparison of Experiments

The following fundamental theorem is due to [Blackwell ’53]:

Theorem 3

Let X, Y be finite spaces. The following are equivalent: (i) Q2 is garbled with respect to Q1. (ii) The information structure induced by channel Q1 is more informative than the one induced by channel Q2 for all single player decision problems with finite U. It is known that i) implies ii) even for standard Borel X, Y, U [Yüksel-Basar ’13]. The converse is significantly more challenging. For the case with general spaces, related results are attributed to [Boll ’55], and [Cartier-Meyer ’64], [Strassen ’65]. In Section 2, will present a direct converse that will be utilized in our main result of the paper and present a brief comparative discussion.

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Zero-Sum Game Setup

Now, we consider an extension of these single-player decision problems to two-player, one-stage, simultaneous-action zero-sum games. We label the two players as Player 1 (the minimizer) and Player 2 (the maximizer). Each player makes a private measurement yi = g(x, νi), where the noise variables v1 and v2 are

• independent. Suppose that gi induces a channel Qi for i = 1, 2.

Given fixed X, Y1, Y2, and ζ such that x ∼ ζ, a game G = (c(x, u1, u2), U1, U2) is a triple of a measurable and bounded cost function c(x, u1, u2) : X × U1 × U2 → R and action spaces for each player U1, U2. Player 1’s goal is to minimize the expected vale of the cost function, and Player 2’s goal is to maximize it. Both players have individual policy spaces Γi := {γi : Yi → Ui}.

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Assumptions

We will impose one of the following conditions on the information structures.

Assumption 1

The information structure is absolutely continuous with respect to a product measure: P(dy1, dy2, dx) ≪ ¯ Q1(dy1)¯ Q2(dy2)P(dx), for reference probability measures ¯ Qi, i = 1, 2. That is, there exists an integrable f which satisfies for every Borel A, B, C P(y1 ∈ B, y2 ∈ C, x ∈ A) =

• A,B,C

f(x, y1, y2)P(dx)¯ Q1(dy1)¯ Q2(dy2)

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Assumptions

Assumption 2

The following conditional independence (or Markov) condition holds: P(dy1, dy2, dx) = Q1(dy1|x)Q2(dy2|x)P(dx) where the measurements of agents are absolutely continuous so that for i = 1, 2, there exists a non-negative function f i and a reference probability measure ¯ Qi such that for all Borel S: Qi(yi ∈ S|x) =

• S

f i(yi, x)¯ Qi(dyi) Note that Assumption 2 implies Assumption 1. We will present the results using Assumption 2 for simpler presentation.

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Information Structures

Let the joint measure µ(dy1, dy2, x) define the information structure. For a zero-sum game with Assumption 2, an information structure µ consists of private information structures µ1 and µ2 defined as the joint probability measure induced on P(X × Yi) by measurement channel Qi with input distribution P(dx) We will allow policies to be randomized with independent randomness. Define the following cost functional: J(P, µ1, γ1, µ2, γ2) = E

Q1,Q2,γ P

• c(x, u1, u2)
• =
• X×Y

c(x, γ1(y1), γ2(y2))Q1(dy1|x)Q2(dy2|x)P(dx) And define Vµ

G(γ1, γ2) as:

Vµ

G(γ1, γ2) =

• c(x, γ1(y1), γ2(y2))µ(dy1, dy2|x)P(dx)

Let V∗(G, µ) be Vµ

G(γ1, γ2) where (γ1, γ2) are chosen to be the equilibrium strategies for the players. 12 / 27

Information Structures

Let the joint measure µ(dy1, dy2, x) define the information structure. For a zero-sum game with Assumption 2, an information structure µ consists of private information structures µ1 and µ2 defined as the joint probability measure induced on P(X × Yi) by measurement channel Qi with input distribution P(dx) We will allow policies to be randomized with independent randomness. Define the following cost functional: J(P, µ1, γ1, µ2, γ2) = E

Q1,Q2,γ P

• c(x, u1, u2)
• =
• X×Y

c(x, γ1(y1), γ2(y2))Q1(dy1|x)Q2(dy2|x)P(dx) And define Vµ

G(γ1, γ2) as:

Vµ

G(γ1, γ2) =

• c(x, γ1(y1), γ2(y2))µ(dy1, dy2|x)P(dx)

Let V∗(G, µ) be Vµ

G(γ1, γ2) where (γ1, γ2) are chosen to be the equilibrium strategies for the players. 12 / 27

Information Structures

Let the joint measure µ(dy1, dy2, x) define the information structure. For a zero-sum game with Assumption 2, an information structure µ consists of private information structures µ1 and µ2 defined as the joint probability measure induced on P(X × Yi) by measurement channel Qi with input distribution P(dx) We will allow policies to be randomized with independent randomness. Define the following cost functional: J(P, µ1, γ1, µ2, γ2) = E

Q1,Q2,γ P

• c(x, u1, u2)
• =
• X×Y

c(x, γ1(y1), γ2(y2))Q1(dy1|x)Q2(dy2|x)P(dx) And define Vµ

G(γ1, γ2) as:

Vµ

G(γ1, γ2) =

• c(x, γ1(y1), γ2(y2))µ(dy1, dy2|x)P(dx)

Let V∗(G, µ) be Vµ

G(γ1, γ2) where (γ1, γ2) are chosen to be the equilibrium strategies for the players. 12 / 27

Information Structures

Let the joint measure µ(dy1, dy2, x) define the information structure. For a zero-sum game with Assumption 2, an information structure µ consists of private information structures µ1 and µ2 defined as the joint probability measure induced on P(X × Yi) by measurement channel Qi with input distribution P(dx) We will allow policies to be randomized with independent randomness. Define the following cost functional: J(P, µ1, γ1, µ2, γ2) = E

Q1,Q2,γ P

• c(x, u1, u2)
• =
• X×Y

c(x, γ1(y1), γ2(y2))Q1(dy1|x)Q2(dy2|x)P(dx) And define Vµ

G(γ1, γ2) as:

Vµ

G(γ1, γ2) =

• c(x, γ1(y1), γ2(y2))µ(dy1, dy2|x)P(dx)

Let V∗(G, µ) be Vµ

G(γ1, γ2) where (γ1, γ2) are chosen to be the equilibrium strategies for the players. 12 / 27

Definitions

Definition 4

Given an information structure µ, we say that γ1,∗, γ2,∗ is an equilibrium for the zero-sum game if inf

γ1∈Γ1 J(P, µ1, γ1, µ2, γ2,∗)

= J(P, µ1, γ1,∗, µ2, γ2,∗) = sup

γ2∈Γ2 J(P, µ1, γ1,∗, µ2, γ2)

Definition 5

We say that an information structure µ is better for the maximizer than information structure Φ, (written as Φ µ) over all games in a set of games G if and only if for all games in G: V∗(G, µ) ≥ V∗(G, Φ)

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Definitions

Definition 4

Given an information structure µ, we say that γ1,∗, γ2,∗ is an equilibrium for the zero-sum game if inf

γ1∈Γ1 J(P, µ1, γ1, µ2, γ2,∗)

= J(P, µ1, γ1,∗, µ2, γ2,∗) = sup

γ2∈Γ2 J(P, µ1, γ1,∗, µ2, γ2)

Definition 5

We say that an information structure µ is better for the maximizer than information structure Φ, (written as Φ µ) over all games in a set of games G if and only if for all games in G: V∗(G, µ) ≥ V∗(G, Φ)

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Garbling

We need to extend the definition of garbling to this case with two players. Here, garbling by a stochastic kernel κi only degrades the information for Player i and leaves the other player’s information unchanged.

Definition 6

We denote by κiµ the information structure in which player i’s information from µ is garbled by a stochastic kernel κi. Explicitly, this means the information structure becomes: (κiµ)(B, dy−i, dx) =

• Yi κi(B|yi)µ(dyi, dy−i, dx), B ∈ B(Yi)

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Comparison in Finite Spaces

The following theorem from [P˛ eski ’08] provides a complete characterization of the comparison of information structures in zero-sum games in the finite case.

Theorem 7 (P˛ eski ’08)

Let X, Y1, Y2 be finite. For any two information structures µ and ψ, µ is better for the maximizer than ψ over all zero-sum games with finite action spaces U1, U2 if and only if there exist kernels κi ∈ Ki, i = min, max, such that κminψ = κmaxµ

15 / 27

Contributions

We will derive a standard Borel counterpart of Theorem 7 characterizing an ordering of information structures for zero-sum games (Theorem 13). We also present two supporting results: [(a)] As a minor technical contribution, we present sufficient conditions for the existence of saddle points in Bayesian zero-sum games with incomplete information in standard Borel spaces (Theorem 8). [(b)] As a further supporting theorem, we will present a partial converse to Blackwell’s ordering theorem for standard Borel spaces, using a separating hyperplane argument and properties of locally convex spaces (Theorem 9). This presents an explicit, self-sufficient derivation for a converse theorem to be utilized in our main theorem, though related comprehensive results have been reported in the literature, as we note.

16 / 27

Contributions

We will derive a standard Borel counterpart of Theorem 7 characterizing an ordering of information structures for zero-sum games (Theorem 13). We also present two supporting results: [(a)] As a minor technical contribution, we present sufficient conditions for the existence of saddle points in Bayesian zero-sum games with incomplete information in standard Borel spaces (Theorem 8). [(b)] As a further supporting theorem, we will present a partial converse to Blackwell’s ordering theorem for standard Borel spaces, using a separating hyperplane argument and properties of locally convex spaces (Theorem 9). This presents an explicit, self-sufficient derivation for a converse theorem to be utilized in our main theorem, though related comprehensive results have been reported in the literature, as we note.

16 / 27

Contributions

We will derive a standard Borel counterpart of Theorem 7 characterizing an ordering of information structures for zero-sum games (Theorem 13). We also present two supporting results: [(a)] As a minor technical contribution, we present sufficient conditions for the existence of saddle points in Bayesian zero-sum games with incomplete information in standard Borel spaces (Theorem 8). [(b)] As a further supporting theorem, we will present a partial converse to Blackwell’s ordering theorem for standard Borel spaces, using a separating hyperplane argument and properties of locally convex spaces (Theorem 9). This presents an explicit, self-sufficient derivation for a converse theorem to be utilized in our main theorem, though related comprehensive results have been reported in the literature, as we note.

16 / 27

Outline

1

Problem Setup

2

Supporting Results

3

Comparison of Information Structures in Zero-Sum Games

17 / 27

Existence of Equilibria

We present sufficient conditions for an equilibrium to exist. Results nearly equivalent to this have been reported, most notably in [Milgrom-Weber ’85]. In a strict sense though, our conditions are more direct and general as stated.

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Existence of Equilibria

Theorem 8 (Existence of Equilibria)

Assume that Assumption 2 holds. Further, let the following hold. (i) The action spaces of players, U1, U2, are compact. (ii) The cost function c is bounded and continuous in players’ actions, for every state of nature x. Then an equilibrium exists under possibly randomized policies, and so there exists a value of the zero-sum game.

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Existence of Equilibria

The proof of this theorem relies on the independent-measurements reduction, essentially due to [Witsenhausen ’88], which is applicable due to Assumption 2. y2 y1 Q1 Q2 ⇐ ⇒ ¯ Q1 ¯ Q2 y1 y2

c(x, u1, u2) ¯ c(x, y1, y2, u1, u2) = c(x, u1, u2)f(x, y1, y2)

u1 u1 u2 u2 γ1 γ1 γ2 γ2 x

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A Partial Converse to Blackwell’s Theorem

Theorem 9

Consider a single DM setup and assume that there exists a function f and a reference probability measure ¯ Q such that for all Borel S: P(y ∈ S|x) =

• S

f(y, x)¯ Q(dy). If Y is compact and an information structure µ is more informative than another information structure Φ over all single-player decision problems with compact standard Borel action spaces and bounded cost functions c(x, u) that are continuous in u for every x, then Φ must be a garbling

• f µ in the sense of Definition 2.

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A Partial Converse to Blackwell’s Theorem

The proof relies on applying the Hahn-Banach Separation Theorem for Locally Convex

• Spaces. By assuming Φ is not a garbling of µ, the separation theorem gives a function that

separates the probability spaces defined by the singleton Φ and the space of all garblings of µ. This separating function is used as a cost function for a game, which leads to a contradiction that µ is more informative than Φ. As stated earlier, the forward direction is known to hold, but an explicit proof to the converse is not available. This result is often attributed to a related theorem by Strassen [Strassen ’65], but the connection is not explicitly made. (The connection can ultimately be shown though, through some additional work).

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Outline

1

Problem Setup

2

Supporting Results

3

Comparison of Information Structures in Zero-Sum Games

23 / 27

Comparison of Information Structures for Zero-Sum Games

We can now extend P˛ eski’s results from Theorem 7.

Definition 10

For fixed X with x ∼ ζ, and fixed Y1, Y2, we define a class of games ˜ Gζ(X, Y1, Y2) to be all games for which the players have compact action spaces and the cost function is bounded and continuous in players’ actions for every state x.

Lemma 11

Given fixed X, ζ, Y1, and Y2, for any information structure µ which satisfies Assumption 1 and any kernels κi ∈ Ki: κmaxµ µ and µ κminµ

• ver all games in ˜

Gζ(X, Y1, Y2).

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Comparison of Information Structures for Zero-Sum Games

We can now extend P˛ eski’s results from Theorem 7.

Definition 10

For fixed X with x ∼ ζ, and fixed Y1, Y2, we define a class of games ˜ Gζ(X, Y1, Y2) to be all games for which the players have compact action spaces and the cost function is bounded and continuous in players’ actions for every state x.

Lemma 11

Given fixed X, ζ, Y1, and Y2, for any information structure µ which satisfies Assumption 1 and any kernels κi ∈ Ki: κmaxµ µ and µ κminµ

• ver all games in ˜

Gζ(X, Y1, Y2).

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Lemma 12

Take fixed X, ζ, fixed and compact Y1, Y2, and information structures Φ and µ which both satisfy Assumption 1. If Φ µ over all games in ˜ Gζ(X, Y1, Y2), then there exist kernels κi ∈ Ki such that: κminΦ = κmaxµ

Theorem 13

Take fixed X, ζ, fixed and compact Y1, Y2, and information structures Φ and µ which both satisfy Assumption 1. Then µ is better for the maximizer than Φ (Φ µ) over all games in ˜ Gζ(X, Y1, Y2) if and only if there exist kernels κi ∈ Ki such that: κminΦ = κmaxµ

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Lemma 12

Take fixed X, ζ, fixed and compact Y1, Y2, and information structures Φ and µ which both satisfy Assumption 1. If Φ µ over all games in ˜ Gζ(X, Y1, Y2), then there exist kernels κi ∈ Ki such that: κminΦ = κmaxµ

Theorem 13

Take fixed X, ζ, fixed and compact Y1, Y2, and information structures Φ and µ which both satisfy Assumption 1. Then µ is better for the maximizer than Φ (Φ µ) over all games in ˜ Gζ(X, Y1, Y2) if and only if there exist kernels κi ∈ Ki such that: κminΦ = κmaxµ

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Meaning of Results

Corollary 14

The value of additional information to a decision maker is never negative for that decision maker in any zero-sum game in ˜ Gζ(X, Y1, Y2). In zero-sum games, improving or hurting both players’ information structures will never give a general benefit to either player over all games. The only time a player will not do worse under a new information structure is if it only makes his channel better, only makes his opponent’s channel worse, makes his channel better and his opponent’s channel worse, or is identical to the previous information structure (and the player is guaranteed to not do worse if any of these conditions holds).

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Meaning of Results

Corollary 14

The value of additional information to a decision maker is never negative for that decision maker in any zero-sum game in ˜ Gζ(X, Y1, Y2). In zero-sum games, improving or hurting both players’ information structures will never give a general benefit to either player over all games. The only time a player will not do worse under a new information structure is if it only makes his channel better, only makes his opponent’s channel worse, makes his channel better and his opponent’s channel worse, or is identical to the previous information structure (and the player is guaranteed to not do worse if any of these conditions holds).

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Conclusion

In this paper, we presented an ordering of information structures for a broad class of zero-sum Bayesian games with incomplete information in standard Borel spaces. We also provided two key supporting results: [i)] a refinement on the conditions for the existence of equilibria in zero-sum games with incomplete information in standard Borel measurement and action spaces and [ii)] a partial converse to Blackwell’s ordering of information structures in this general setting.

27 / 27

Conclusion

In this paper, we presented an ordering of information structures for a broad class of zero-sum Bayesian games with incomplete information in standard Borel spaces. We also provided two key supporting results: [i)] a refinement on the conditions for the existence of equilibria in zero-sum games with incomplete information in standard Borel measurement and action spaces and [ii)] a partial converse to Blackwell’s ordering of information structures in this general setting.

27 / 27

Conclusion

In this paper, we presented an ordering of information structures for a broad class of zero-sum Bayesian games with incomplete information in standard Borel spaces. We also provided two key supporting results: [i)] a refinement on the conditions for the existence of equilibria in zero-sum games with incomplete information in standard Borel measurement and action spaces and [ii)] a partial converse to Blackwell’s ordering of information structures in this general setting.

27 / 27