We prove that in a general zero-sum repeated game where the first player is more informed than th... more We prove that in a general zero-sum repeated game where the first player is more informed than the second player and controls the evolution of information on the state, the uniform value exists. This result extends previous results on Markov decision processes with partial observation (Rosenberg, Solan, Vieille [11]), and repeated games with an informed controller (Renault [10]). Our formal definition of a more informed player is more general than the inclusion of signals, allowing therefore for imperfect monitoring of actions. We construct an auxiliary stochastic game whose state space is the set of second order beliefs of player 2 (beliefs about beliefs of player 1 on the true state variable of the initial game) with perfect monitoring and we prove it has a value by using a result of Renault [10]. A key element in this work is to prove that player 1 can use strategies of the auxiliary game in the initial game in our general framework, which allows to deduce that the value of the auxiliary game is also the value of our initial repeated game by using classical arguments.
Differential games with asymmetric information were introduced by Cardaliaguet (2007). As in repe... more Differential games with asymmetric information were introduced by Cardaliaguet (2007). As in repeated games with lack of information on both sides (Aumann and Maschler (1995)), each player receives a private signal (his type) before the game starts and has a prior belief about his opponent's type. Then, a differential game is played in which the dynamic and the payoff function depend on both types: each player is thus partially informed about the differential game that is played. The existence of the value function and some characterizations have been obtained under the assumption that the signals are drawn independently. In this paper, we drop this assumption and extend these two results to the general case of correlated types. This result is then applied to repeated games with incomplete information: the characterization of the asymptotic value obtained by Rosenberg and Sorin (2001) and Laraki (2001) for the independent case is extended to the general case.
In an optimal control framework, we consider the value V T (x) of the problem starting from state... more In an optimal control framework, we consider the value V T (x) of the problem starting from state x with finite horizon T , as well as the value V λ (x) of the λ-discounted problem starting from x. We prove that uniform convergence (on the set of states) of the values V T (•) as T tends to infinity is equivalent to uniform convergence of the values V λ (•) as λ tends to 0, and that the limits are identical. An example is also provided to show that the result does not hold for pointwise convergence. This work is an extension, using similar techniques, of a related result in a discrete-time framework [15].
The value of a zero-sum dierential games is known to exist, under Isaacs condition, as the unique... more The value of a zero-sum dierential games is known to exist, under Isaacs condition, as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation. In this note we provide a new proof via the construction of ε-optimal strategies, which is inspired in the extremal aiming method from [3].
In the COVID-19 pandemic, governments have, among other measures,1,2 mandated the use of COVID ce... more In the COVID-19 pandemic, governments have, among other measures,1,2 mandated the use of COVID certificates to prove vaccination, recovery, or a recent negative test, and have required them to access shops, restaurants, schools, universities, or workplaces.3 While arguments for and against COVID certificates have focused on reducing transmission and ethical concerns,4,5 the incentive effect of COVID certificates on vaccine uptake, health outcomes, and the economy has not yet been investigated. To estimate this effect, we construct counterfactuals based on innovation diffusion theory6 for France, Germany, and Italy. We estimate that the announcement of COVID certificates during summer 2021 led to increased vaccine uptake in France of 13.0 (95% CI 9.7–14.9) percentage points (p.p.) of the total population up to the end of the year, in Germany 6.2 (2.6–6.9) p.p., and in Italy 9.7 (5.4–12.3) p.p. Further, this averted an additional 3,979 (3,453–4,298) deaths in France, 1,133 (-312–1,358...
Abstract: In an optimal control framework, we consider the value VT (x) of the problem starting f... more Abstract: In an optimal control framework, we consider the value VT (x) of the problem starting from state x with finite horizon T, as well as the value Wλ(x) of the λ-discounted problem starting from x. We prove that uniform convergence (on the set of states) of the values VT (·) as T tends to infinity is equivalent to uniform convergence of the values Wλ(·) as λ tends to 0, and that the limits are identical. An example is also provided to show that the result does not hold for pointwise convergence. This work is an extension, using similar techniques, of a related result by Lehrer and Sorin in a discrete-time framework. 1
Authors are encouraged to submit new papers to INFORMS journals by means of a style file template... more Authors are encouraged to submit new papers to INFORMS journals by means of a style file template, which includes the journal title. However, use of a template does not certify that the paper has been accepted for publication in the named journal. INFORMS journal templates are for the exclusive purpose of submitting to an INFORMS journal and should not be used to distribute the papers in print or online or to submit the papers to another publication.
We provide a direct, elementary proof for the existence of lim λ→0 v λ , where v λ is the value o... more We provide a direct, elementary proof for the existence of lim λ→0 v λ , where v λ is the value of λ-discounted finite two-person zero-sum stochastic game. 1 Introduction Two-person zero-sum stochastic games were introduced by Shapley [4]. They are described by a 5-tuple (Ω, I, J , q, g), where Ω is a finite set of states, I and J are finite sets of actions, g : Ω × I × J → [0, 1] is the payoff, q : Ω × I × J → ∆(Ω) the transition and, for any finite set X, ∆(X) denotes the set of probability distributions over X. The functions g and q are bilinearly extended to Ω × ∆(I) × ∆(J). The stochastic game with initial state ω ∈ Ω and discount factor λ ∈ (0, 1] is denoted by Γ λ (ω) and is played as follows: at stage m ≥ 1, knowing the current state ω m , the players choose actions (i m , j m) ∈ I × J ; their choice produces a stage payoff g(ω m , i m , j m) and influences the transition: a new state ω m+1 is chosen according to the probability distribution q(•|ω m , i m , j m). At the end of the game, player 1 receives m≥1 λ(1 − λ) m−1 g(ω m , i m , j m) from player 2. The game Γ λ (ω) has a value v λ (ω), and v λ = (v λ (ω)) ω∈Ω is the unique fixed point of the so-called Shapley operator [4], i.e. v λ = Φ(λ, v λ), where for all f ∈ R Ω :
Using the duality techniques introduced by De Meyer (1996a, 1996b), De Meyer and Marino (2005) pr... more Using the duality techniques introduced by De Meyer (1996a, 1996b), De Meyer and Marino (2005) provided optimal strategies for both players in finitely repeated games with incomplete information on two sides, in the independent case. In this note, we extend both the duality techniques and the construction of optimal strategies to the case of general type dependence.
We introduce the dependent splitting game, a zero-sum stochastic game in which the players jointl... more We introduce the dependent splitting game, a zero-sum stochastic game in which the players jointly control a martingale. This game models the transmission of information in repeated games with incomplete information on both sides, in the dependent case: The state variable represents the martingale of posterior beliefs. We establish the existence of the value for any fixed, general evaluation of the stage payoffs, as a function of the initial state. We then prove the convergence of the value functions, as the evaluation vanishes, to the unique solution of the Mertens–Zamir system of equations is established. From this result, we derive the convergence of the values of repeated games with incomplete information on both sides, in the dependent case, to the same function, as the evaluation vanishes. Finally, we establish a surprising result: Unlike repeated games with incomplete information on both sides, the splitting game has a uniform value. Moreover, we exhibit a couple of optimal s...
We establish, for the first time, a connection between stochastic games and multiparameter eigenv... more We establish, for the first time, a connection between stochastic games and multiparameter eigenvalue problems, using the theory developed by Shapley and Snow (1950). This connection provides new results, new proofs, and new tools for studying stochastic games.
Stochastic games are a classical model in game theory in which two opponents interact and the env... more Stochastic games are a classical model in game theory in which two opponents interact and the environment changes in response to the players' behavior. The central solution concepts for these games are the discounted values and the value, which represent what playing the game is worth to the players for different levels of impatience. In the present manuscript, we provide algorithms for computing exact expressions for the discounted values and for the value, which are polynomial in the number of pure stationary strategies of the players. This result considerably improves all the existing algorithms, including the most efficient one, due to Hansen, Kouck\'y, Lauritzen, Miltersen and Tsigaridas (STOC 2011).
The value of a zero-sum differential games is known to exist, under Isaacs' condition, as the... more The value of a zero-sum differential games is known to exist, under Isaacs' condition, as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation. In this note we provide a self-contained proof based on the construction of $\ep$-optimal strategies, which is inspired by the "extremal aiming" method from Krasovskii and Subbotin.
In a zero-sum stochastic game, at each stage, two adversary players take decisions and receive a ... more In a zero-sum stochastic game, at each stage, two adversary players take decisions and receive a stage payoff determined by them and by a random variable representing the state of nature. The total payoff is the discounted sum of the stage payoffs. Assume that the players are very patient and use optimal strategies. We then prove that, at any point in the game, players get essentially the same expected payoff: the payoff is constant. This solves a conjecture by Sorin, Venel and Vigeral (2010). The proof relies on the semi-algebraic approach for discounted stochastic games introduced by Bewley and Kohlberg (1976), on the theory of Markov chains with rare transitions, initiated by Friedlin and Wentzell (1984), and on some variational inequalities for value functions inspired by the recent work of Davini, Fathi, Iturriaga and Zavidovique (2016).
We establish, for the first time, a connection between stochastic games and multi-parameter eigen... more We establish, for the first time, a connection between stochastic games and multi-parameter eigenvalue problems, using the theory developed by Shapley and Snow (1950) in the context of matrix games. This connection provides new results, new proofs, and new tools for studying stochastic games.
Stochastic games are a classical model in game theory in which two opponents interact over a fini... more Stochastic games are a classical model in game theory in which two opponents interact over a finite set of states that change stochastically over time. The two central solution concepts for these games are the discounted values and the limit values. Efficiently computing these values is an important problem in game theory. In this paper, we describe new algorithms for computing them, which run in polynomial time for any fixed number of states, and which considerably improve the dependence on the number of states given by Hansen, Koucky, Lauritzen, Miltersen and Tsigaridas (STOC 2011).
In this dissertation we study several aspects of two-player zero-sum games. Morespecifically, we ... more In this dissertation we study several aspects of two-player zero-sum games. Morespecifically, we are concerned with problems related to information and dynamics, both indiscrete and continuous time games. The manuscript is divided in two parts. The first part is devoted to discrete-time models such as stochastic games, introduced by Shapley (1953), repeated games with incomplete information on two sides, first studied by Aumann, Maschler and Stearns (1995, originally in 1967-68) and the general model of repeated games proposed by Mertens, Sorin et Zamir (1994). First, we provide a new proof of the convergence of the discounted values of stochastic games. Then, in the context of games with incomplete information, we extend the duality techniques of De Meyer (1996) to games where the players’ information is correlated. This result is used to establish exact recursive formulae in the two dual games, one for each player, in repeated games with incomplete information. One deduces then th...
We prove that in a general zero-sum repeated game where the first player is more informed than th... more We prove that in a general zero-sum repeated game where the first player is more informed than the second player and controls the evolution of information on the state, the uniform value exists. This result extends previous results on Markov decision processes with partial observation (Rosenberg, Solan, Vieille [11]), and repeated games with an informed controller (Renault [10]). Our formal definition of a more informed player is more general than the inclusion of signals, allowing therefore for imperfect monitoring of actions. We construct an auxiliary stochastic game whose state space is the set of second order beliefs of player 2 (beliefs about beliefs of player 1 on the true state variable of the initial game) with perfect monitoring and we prove it has a value by using a result of Renault [10]. A key element in this work is to prove that player 1 can use strategies of the auxiliary game in the initial game in our general framework, which allows to deduce that the value of the auxiliary game is also the value of our initial repeated game by using classical arguments.
Differential games with asymmetric information were introduced by Cardaliaguet (2007). As in repe... more Differential games with asymmetric information were introduced by Cardaliaguet (2007). As in repeated games with lack of information on both sides (Aumann and Maschler (1995)), each player receives a private signal (his type) before the game starts and has a prior belief about his opponent's type. Then, a differential game is played in which the dynamic and the payoff function depend on both types: each player is thus partially informed about the differential game that is played. The existence of the value function and some characterizations have been obtained under the assumption that the signals are drawn independently. In this paper, we drop this assumption and extend these two results to the general case of correlated types. This result is then applied to repeated games with incomplete information: the characterization of the asymptotic value obtained by Rosenberg and Sorin (2001) and Laraki (2001) for the independent case is extended to the general case.
In an optimal control framework, we consider the value V T (x) of the problem starting from state... more In an optimal control framework, we consider the value V T (x) of the problem starting from state x with finite horizon T , as well as the value V λ (x) of the λ-discounted problem starting from x. We prove that uniform convergence (on the set of states) of the values V T (•) as T tends to infinity is equivalent to uniform convergence of the values V λ (•) as λ tends to 0, and that the limits are identical. An example is also provided to show that the result does not hold for pointwise convergence. This work is an extension, using similar techniques, of a related result in a discrete-time framework [15].
The value of a zero-sum dierential games is known to exist, under Isaacs condition, as the unique... more The value of a zero-sum dierential games is known to exist, under Isaacs condition, as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation. In this note we provide a new proof via the construction of ε-optimal strategies, which is inspired in the extremal aiming method from [3].
In the COVID-19 pandemic, governments have, among other measures,1,2 mandated the use of COVID ce... more In the COVID-19 pandemic, governments have, among other measures,1,2 mandated the use of COVID certificates to prove vaccination, recovery, or a recent negative test, and have required them to access shops, restaurants, schools, universities, or workplaces.3 While arguments for and against COVID certificates have focused on reducing transmission and ethical concerns,4,5 the incentive effect of COVID certificates on vaccine uptake, health outcomes, and the economy has not yet been investigated. To estimate this effect, we construct counterfactuals based on innovation diffusion theory6 for France, Germany, and Italy. We estimate that the announcement of COVID certificates during summer 2021 led to increased vaccine uptake in France of 13.0 (95% CI 9.7–14.9) percentage points (p.p.) of the total population up to the end of the year, in Germany 6.2 (2.6–6.9) p.p., and in Italy 9.7 (5.4–12.3) p.p. Further, this averted an additional 3,979 (3,453–4,298) deaths in France, 1,133 (-312–1,358...
Abstract: In an optimal control framework, we consider the value VT (x) of the problem starting f... more Abstract: In an optimal control framework, we consider the value VT (x) of the problem starting from state x with finite horizon T, as well as the value Wλ(x) of the λ-discounted problem starting from x. We prove that uniform convergence (on the set of states) of the values VT (·) as T tends to infinity is equivalent to uniform convergence of the values Wλ(·) as λ tends to 0, and that the limits are identical. An example is also provided to show that the result does not hold for pointwise convergence. This work is an extension, using similar techniques, of a related result by Lehrer and Sorin in a discrete-time framework. 1
Authors are encouraged to submit new papers to INFORMS journals by means of a style file template... more Authors are encouraged to submit new papers to INFORMS journals by means of a style file template, which includes the journal title. However, use of a template does not certify that the paper has been accepted for publication in the named journal. INFORMS journal templates are for the exclusive purpose of submitting to an INFORMS journal and should not be used to distribute the papers in print or online or to submit the papers to another publication.
We provide a direct, elementary proof for the existence of lim λ→0 v λ , where v λ is the value o... more We provide a direct, elementary proof for the existence of lim λ→0 v λ , where v λ is the value of λ-discounted finite two-person zero-sum stochastic game. 1 Introduction Two-person zero-sum stochastic games were introduced by Shapley [4]. They are described by a 5-tuple (Ω, I, J , q, g), where Ω is a finite set of states, I and J are finite sets of actions, g : Ω × I × J → [0, 1] is the payoff, q : Ω × I × J → ∆(Ω) the transition and, for any finite set X, ∆(X) denotes the set of probability distributions over X. The functions g and q are bilinearly extended to Ω × ∆(I) × ∆(J). The stochastic game with initial state ω ∈ Ω and discount factor λ ∈ (0, 1] is denoted by Γ λ (ω) and is played as follows: at stage m ≥ 1, knowing the current state ω m , the players choose actions (i m , j m) ∈ I × J ; their choice produces a stage payoff g(ω m , i m , j m) and influences the transition: a new state ω m+1 is chosen according to the probability distribution q(•|ω m , i m , j m). At the end of the game, player 1 receives m≥1 λ(1 − λ) m−1 g(ω m , i m , j m) from player 2. The game Γ λ (ω) has a value v λ (ω), and v λ = (v λ (ω)) ω∈Ω is the unique fixed point of the so-called Shapley operator [4], i.e. v λ = Φ(λ, v λ), where for all f ∈ R Ω :
Using the duality techniques introduced by De Meyer (1996a, 1996b), De Meyer and Marino (2005) pr... more Using the duality techniques introduced by De Meyer (1996a, 1996b), De Meyer and Marino (2005) provided optimal strategies for both players in finitely repeated games with incomplete information on two sides, in the independent case. In this note, we extend both the duality techniques and the construction of optimal strategies to the case of general type dependence.
We introduce the dependent splitting game, a zero-sum stochastic game in which the players jointl... more We introduce the dependent splitting game, a zero-sum stochastic game in which the players jointly control a martingale. This game models the transmission of information in repeated games with incomplete information on both sides, in the dependent case: The state variable represents the martingale of posterior beliefs. We establish the existence of the value for any fixed, general evaluation of the stage payoffs, as a function of the initial state. We then prove the convergence of the value functions, as the evaluation vanishes, to the unique solution of the Mertens–Zamir system of equations is established. From this result, we derive the convergence of the values of repeated games with incomplete information on both sides, in the dependent case, to the same function, as the evaluation vanishes. Finally, we establish a surprising result: Unlike repeated games with incomplete information on both sides, the splitting game has a uniform value. Moreover, we exhibit a couple of optimal s...
We establish, for the first time, a connection between stochastic games and multiparameter eigenv... more We establish, for the first time, a connection between stochastic games and multiparameter eigenvalue problems, using the theory developed by Shapley and Snow (1950). This connection provides new results, new proofs, and new tools for studying stochastic games.
Stochastic games are a classical model in game theory in which two opponents interact and the env... more Stochastic games are a classical model in game theory in which two opponents interact and the environment changes in response to the players' behavior. The central solution concepts for these games are the discounted values and the value, which represent what playing the game is worth to the players for different levels of impatience. In the present manuscript, we provide algorithms for computing exact expressions for the discounted values and for the value, which are polynomial in the number of pure stationary strategies of the players. This result considerably improves all the existing algorithms, including the most efficient one, due to Hansen, Kouck\'y, Lauritzen, Miltersen and Tsigaridas (STOC 2011).
The value of a zero-sum differential games is known to exist, under Isaacs' condition, as the... more The value of a zero-sum differential games is known to exist, under Isaacs' condition, as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation. In this note we provide a self-contained proof based on the construction of $\ep$-optimal strategies, which is inspired by the "extremal aiming" method from Krasovskii and Subbotin.
In a zero-sum stochastic game, at each stage, two adversary players take decisions and receive a ... more In a zero-sum stochastic game, at each stage, two adversary players take decisions and receive a stage payoff determined by them and by a random variable representing the state of nature. The total payoff is the discounted sum of the stage payoffs. Assume that the players are very patient and use optimal strategies. We then prove that, at any point in the game, players get essentially the same expected payoff: the payoff is constant. This solves a conjecture by Sorin, Venel and Vigeral (2010). The proof relies on the semi-algebraic approach for discounted stochastic games introduced by Bewley and Kohlberg (1976), on the theory of Markov chains with rare transitions, initiated by Friedlin and Wentzell (1984), and on some variational inequalities for value functions inspired by the recent work of Davini, Fathi, Iturriaga and Zavidovique (2016).
We establish, for the first time, a connection between stochastic games and multi-parameter eigen... more We establish, for the first time, a connection between stochastic games and multi-parameter eigenvalue problems, using the theory developed by Shapley and Snow (1950) in the context of matrix games. This connection provides new results, new proofs, and new tools for studying stochastic games.
Stochastic games are a classical model in game theory in which two opponents interact over a fini... more Stochastic games are a classical model in game theory in which two opponents interact over a finite set of states that change stochastically over time. The two central solution concepts for these games are the discounted values and the limit values. Efficiently computing these values is an important problem in game theory. In this paper, we describe new algorithms for computing them, which run in polynomial time for any fixed number of states, and which considerably improve the dependence on the number of states given by Hansen, Koucky, Lauritzen, Miltersen and Tsigaridas (STOC 2011).
In this dissertation we study several aspects of two-player zero-sum games. Morespecifically, we ... more In this dissertation we study several aspects of two-player zero-sum games. Morespecifically, we are concerned with problems related to information and dynamics, both indiscrete and continuous time games. The manuscript is divided in two parts. The first part is devoted to discrete-time models such as stochastic games, introduced by Shapley (1953), repeated games with incomplete information on two sides, first studied by Aumann, Maschler and Stearns (1995, originally in 1967-68) and the general model of repeated games proposed by Mertens, Sorin et Zamir (1994). First, we provide a new proof of the convergence of the discounted values of stochastic games. Then, in the context of games with incomplete information, we extend the duality techniques of De Meyer (1996) to games where the players’ information is correlated. This result is used to establish exact recursive formulae in the two dual games, one for each player, in repeated games with incomplete information. One deduces then th...
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Papers by Miquel Barton