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 Problem Setup 1 Supporting Results 2 Comparison of Information Structures in Zero-Sum Games 3 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. 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. 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
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 ) = E Q ,γ [ c ( x , γ ( y )] P 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
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