Haut PDF On the encoding and solving partial information games

On the encoding and solving partial information games

On the encoding and solving partial information games

to solve regarding games in our context is the existence of a winning strategy for the player modeling the system. This is now well understood. We know that total information parity games enjoy the memoryless determinacy property [18] ensuring that in each game, one of the players has a winning strategy, and that a winning strategy exists if and only if there is a memoryless winning strategy, i.e. a strategy that depends only on the last visited node of the graph, and not on a history of the play. However, partial information games do not enjoy this property since the player may need memory to win the game. On the other hand, regarding tools implementations, the field of two-player games has not reached the maturity obtained in model-checkers area. For total information games, to the notable exception of pgsolver [24] that provides a platform of implementa- tion for algorithms solving parity games, and Uppaal-TiGa[30] that solves in a very efficient way timed games (but restricted to reachability conditions), few im- plementations are available. SAT-implementations of restricted types of games have also been proposed [17], as well as a reduction of parity games to SAT [21]. As for partial information games, even less attempts have been made. To our knowledge, only alpaga [1] solves partial information games, but the explicit input format does not allow to solve real-life instances.
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Cooperative Games with Incomplete Information : Some Open Problems

Cooperative Games with Incomplete Information : Some Open Problems

procedure in a game in extensive form, which accounts for the players’ nego- tiation possibilities. As a topic for further research, Myerson (1984b)’s value should be further investigated and challenged. The previous paragraphs illustrate that, even in the absence of strategic externalities, the axiomatic approach to cooperation, which was so fruit- ful under complete information, to date renders much less clear conclusions when negotiation takes place between privately informed players. As recalled above, Myerson (1984a) proposes a partial axiomatization of a bargaining solution in the case of two equally powerful players. Focusing on the issue of information revelation at the negotiation stage, de Clippel and Minelli (2004) pursue this analysis. They provide cooperative and noncooperative characterizations of Myerson (1983, 1984a)’s solutions under the additional assumption that types become verifiable at the stage where a mechanism is used to make decisions. de Clippel and Minelli (2004) propose in particular a refinement of Wilson (1978)’s coarse core. An obvious open problem is the extension of these results when types remain unverifiable, even at the decision stage. In any case, this contribution, as other recent ones, e.g., Serrano and Vohra (2007), detailed in section 2, and de Clippel (2005), indicates that the analysis of simple, explicit negotiation procedures is a promising approach given the state of the art.
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Single-player games: introduction to a new solving method

Single-player games: introduction to a new solving method

The low-level is a set of algorithms for solving subproblems of the game. Classical state-space algorithms can be used but not exclusively. We believe that it is possible for almost all games to determine primitive game elements that have to reach some goal. In puzzles like the 24-tile puzzle, the agents could be defined as the tiles. In this representation, each tile aims to reach its final destination but cannot move without altering the position of other agents. In the game of Sokoban, all the agents are instances of stones of the maze and have thus the same characteristics. For other games however, we could define agents that have their own personality. For the game of solitaire for example, the agents could be the 52 cards. Each agent is now unique. Note that for such imperfect information game, we must consider that only a subset of the agents is visible. The other agents can thus be seen as being in an unknown queue, waiting for entering into play.
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Solving Simple Stochastic Tail Games

Solving Simple Stochastic Tail Games

In this paper, we focus our attention on simple stochastic tail games. By “simple” we mean that players have perfect-information and take their decisions turn-by-turn, whereas in stochastic games as introduced initially by Shapley [Sha53] players take their decisions concurrently. A tail winning condition is such that the winner of a play does not depend on finite prefixes of the play, only the long-term behaviour of the play matters. This class encompasses games for verification and mean-payoff games. From a verification perspective, tail conditions correspond to cases where local glitches are tolerated in the beginning of a run, as long as the specification is met in the long-run, e.g. in self-stabilising protocols.
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Cooperative Games with Incomplete Information : Some Open Problems

Cooperative Games with Incomplete Information : Some Open Problems

procedure in a game in extensive form, which accounts for the players’ nego- tiation possibilities. As a topic for further research, Myerson (1984b)’s value should be further investigated and challenged. The previous paragraphs illustrate that, even in the absence of strategic externalities, the axiomatic approach to cooperation, which was so fruit- ful under complete information, to date renders much less clear conclusions when negotiation takes place between privately informed players. As recalled above, Myerson (1984a) proposes a partial axiomatization of a bargaining solution in the case of two equally powerful players. Focusing on the issue of information revelation at the negotiation stage, de Clippel and Minelli (2004) pursue this analysis. They provide cooperative and noncooperative characterizations of Myerson (1983, 1984a)’s solutions under the additional assumption that types become verifiable at the stage where a mechanism is used to make decisions. de Clippel and Minelli (2004) propose in particular a refinement of Wilson (1978)’s coarse core. An obvious open problem is the extension of these results when types remain unverifiable, even at the decision stage. In any case, this contribution, as other recent ones, e.g., Serrano and Vohra (2007), detailed in section 2, and de Clippel (2005), indicates that the analysis of simple, explicit negotiation procedures is a promising approach given the state of the art.
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Pure Strategies in Imperfect Information Stochastic Games

Pure Strategies in Imperfect Information Stochastic Games

value corresponding to the bit of the same index in the incremented version of the described number (this can be computed on the fly). This marking by Adam is done by him playing a distinguished action; the checking is done deterministically (thanks to a counter). One also uses this binary encoding of the index of the cell in the following way: whenever Adam marks a symbol that he claims will be incorrectly updated in the next configuration, a bit of its binary encoding is guessed (i.e. randomly chosen) and its index is stored and not observed by none of the players. Later, when Adam indicates the supposed corresponding symbol in the next configuration, the guessed bit is checked and should match: if not the play goes to a final state and Eve wins; otherwise one does as previously explained (i.e. one checks whether the symbol is correct: if not the play restarts otherwise the play goes to a final state and Eve wins). Hence, in the game’s state one also stores (and hides to both players) the value and index of the randomly chosen bit.
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Representing and Solving Hedonic Games with Ordinal Preferences and Thresholds

Representing and Solving Hedonic Games with Ordinal Preferences and Thresholds

agents i would like to see in her coalitions and which agents she would like not to: For instance, if  1 is 2  1 3  1 4, we know that 1 prefers 2 to 3 and 3 to 4, but nothing tells us whether 1 prefers to be with 2 (respectively, 3 and 4) to being alone, that is, if the abso- lute desirability of 2, 3, and 4 is positive or negative (of course, if it is negative for 3, it is also negative for 4, etc.). So, both ways are insufficiently informative: Specifying only a partition into positive and negative agents (“friends” and “enemies”) does not tell which of her friends i prefers to which other agents, and which of her en- emies she wants to avoid most. On the other hand, specifying a ranking over agents does not say which agents i prefers to be with rather than being alone. Here we propose a model that integrates the models of Cases 1, 3, and 4: Each agent i first subdivides the other agents into three groups, her friends, her enemies, and an in- termediate type of agents on which she has neither a positive nor a negative opinion and then specifies a ranking of her friends and enemies. Based on this representation, we consider a natural exten- sion to a player’s preference, the generalized Bossong–Schweigert extension (see [8, 14]), which is a partial order over coalitions con- taining the player. A related model can be found in the context of matching theory: Responsive preferences are studied in bipar- tite many-to-one matching markets and consider the comparison of one participant to another, 1 although not in distinction of friends or enemies (see, e.g., [19, 20]). In the following, we consider differ- ent ways of how to deal with incomparabilities within these partial orders. A first approach is to leave incomparabilities open and de- fine notions such as “possible” and “necessary” stability concepts. A second approach is to define comparability functions in order to determine the relation between incomparable coalitions that extend
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Possibilistic Games with Incomplete Information

Possibilistic Games with Incomplete Information

vary the number of players from 2 to 6, the number of types from 2 to 8 and the number of actions from 2 to 15. For each combination of parameters, we have generated 50 instances and measured the time necessary to get a Π-NE (or a neg- ative result). We present in the following results of 3 game classes: Covariant games, Dispersion games and Travelers Dilemma game. All experiments were conducted on an In- tel Xeon E5540 processor and 64GB RAM workstation. We used CPLEX [CPLEX, 2009] as a MILP solver. We also im- plemented the transformation of the Π-game as a normal form game (T ˜ G) in Java 8. This method, which is exponential in time and space, cannot be considered as a solving method, and this is supported by the experimental results. The im- plementation of the T ˜ G and MILP solver are available online [Ben Amor et al., 2019]. In our evaluation, we bounded the execution time to 10 minutes as in [Sandholm et al., 2005; Porter et al., 2008] experiments.
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Qualitative Concurrent Stochastic Games with Imperfect Information

Qualitative Concurrent Stochastic Games with Imperfect Information

In this paper, we remove those two restrictions by considering concurrent stochastic games with imperfect information. Those are finite states games in which, at each round, the two players choose simultaneously and independently an action. Then a successor state is chosen accordingly to some fixed probability distribution depending on the previous state and on the pair of actions chosen by the players. Imperfect information is modeled as follows: both players have an equivalence relation over states and, instead of observing the exact state, they only see to which equivalence class it belongs. Therefore, if two partial plays are indistinguishable by some player, he should behave the same in both of them. Note that this model naturally captures several model studied in the literature [1, 11, 7, 8]. The winning conditions we consider here are reachability (is there a final state eventually visited?), B¨ uchi (is there a final state that is visited infinitely often?) and their dual versions, safety and co-B¨ uchi.
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Solving strong Stackelberg equilibrium in stochastic games

Solving strong Stackelberg equilibrium in stochastic games

I. I NTRODUCTION In this work we face the problem of computing a strong Stackelberg equilibrium (SSE) in a stochastic game (SG). Given a set of states we model a two player perfect information dynamic where one of them, called Leader or player A, ob- serves the current state and decides, possible up to probability distribution f , between a set of available actions. Then other player, called Follower or player B, observes the strategy of player A and plays his best response noted by g. We represent a two-person stochastic discrete game G by

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Admissibility in Games with Imperfect Information

Admissibility in Games with Imperfect Information

Digital Object Identifier 10.4230/LIPIcs... 1 Introduction Two-player zero-sum perfect information games played on finite (directed) graphs are the canonical model to formalize the reactive synthesis problem [24, 1]. Unfortunately, this mathematical model is often a too coarse abstraction of reality. First, realistic systems are usually made up of several components, each of them with its own objective. These objectives are not necessarily antagonistic. Hence, the setting of non-zero sum graph games needs to be investigated, see [9] and additional references therein. Second, in systems made of several components, each component has usually a partial view on the entire system. Hence it is natural to study games with imperfect information [25, 13]. In this paper, we investigate the notion of admissible strategies for infinite duration non-zero sum games played on graphs in which players have imperfect information.
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Solving Games and All That

Solving Games and All That

As a subset of general zero-sum imperfect information games, stacked matrix games can be solved by general techniques such as creating a single-matrix game in which individual moves represent pure strategies in the original game. However, because this transformation leads to an exponential blowup, it can only be applied to tiny problems. In their landmark paper, [77] define the sequence form game representation which avoids redundancies present in above game transformation and reduces the game value computation time to polynomial in the game tree size. In the experimental section we present data showing that even for small stacked matrix games, the sequence form approach requires lots of memory and therefore can’t solve larger problems. The main reason is that the algorithm doesn’t detect the regular information set structure present in stacked matrix games, and also computes mixed strategies for all information sets, which may not be necessary. To overcome these problems [56] introduce a loss-less abstraction for games with certain regularity constraints and show that Nash equilibria found in the often much smaller game abstractions correspond to ones in the original game. General stacked matrix games don’t fall into the game class considered in this paper, but the general idea of pre- processing games to transform them into smaller, equivalent ones may also apply to stacked matrix games.
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Acquisition Games with Partial-Asymmetric Information

Acquisition Games with Partial-Asymmetric Information

C ONCLUSIONS We considered lock acquisition games with partial, asym- metric information. Agents attempt to control the rate of their Poisson clocks to acquire two locks, the first one to get both would get the reward. There is a deadline before which the locks are to be acquired, only the first agent to contact the lock can acquire it and the agents are not aware of the acquisition status of others. It is possible that an agent continues its acquisition attempts, while the lock is already acquired by another agent. The agents pay a cost proportional to their rates of acquisition. We proposed a new approach to solve these asymmetric and non-classical information games, ”open loop control till the information update”. With this approach we have dynamic programming equations applicable at state change update instances and then each stage of the dynamic programming equations is to be solved by optimal control theory based tools (HJB equations). We showed that a pair of (available) state dependent time threshold policies form Nash equilibrium. We also conjectured the results for the games with N -agents.
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Ordinal Polymatrix Games with Incomplete Information

Ordinal Polymatrix Games with Incomplete Information

The second part of experiments was dedicated to polyma- trix Π-games. We generated coordination games with dif- ferent numbers of players. We varied the number of players from 5 to 80. Then, we generated random interactions be- tween players (ensuring that the interaction graph was con- nected). Then, for each edge, i.e., interaction, we generated a Π-game between 2 players using the Π-game generator proposed in (Ben Amor et al. 2019b). We varied the number of types from 2 to 9 and the number of actions (for MEG) from 2 to 10. Then we computed the average execution time needed to find a PNE by transforming the original game into its equivalent min-based polymatrix game (using Def- inition 15) and solving the MILP of the latter. Notice that, the equivalent min-based polymatrix game contains n · t 2
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Perfect Information Stochastic Priority Games

Perfect Information Stochastic Priority Games

1 Introduction Recently de Alfaro, Henzinger and Majumdar[4] introduced a new variant of µ- calculus: discounted µ-calculus. As it is known since the seminal paper [6] of Emerson and Jutla µ-calculus is strongly related to parity games and this relationship is pre- served even for stochastic games, [5]. In this context it is natural to ask if there is a class of games that corresponds to discounted µ-calculus of [4]. A partial answer to this question was given in [8], where an appropriate class of infinite discounted games was introduced. However, in [8], only deterministic systems were considered and much more challenging problem of stochastic games was left open. In the present paper we return to the problem but in the context perfect information stochastic games. The most basic and usually non-trivial question is if the games that we consider admit “simple” optimal strategies for both players. We give a positive answer, for all games presented in this paper both players have pure stationary optimal strategies. Since our games contain parity games as a very special case, our paper extends the result known for perfect information parity games [2, 10, 3, 14].
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Evaluating information in zero-sum games with incomplete information on both sides

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

The result concerning games with one-sided information is certainly relevant to the current two-sided information case. In the one-sided and deterministic informa- tion 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 mean- ings: 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.
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Optimal routing among ./M/1 queues with partial information

Optimal routing among ./M/1 queues with partial information

Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex France Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vi[r]

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Essays on two-player games with asymmetric information

Essays on two-player games with asymmetric information

price dynamics are nearly constant. This result is completely different from De Meyer’s (2010). In the N −stage repeated zero-sum game, in each stage, player 2 makes a lottery involving two prices such that the conditional expectation of the price equals the value L. Playing in this way entails a negative price, which is not a natural interpretation in economics. However, we cannot impose a restriction requiring a positive price because this would violate the invariance axiom of the natural trading mechanism. That is, the value of the game must remain unchanged if one shifts the liquidation value L by a constant amount. This result might be improved in the event that player 2 is risk averse. Therefore, in section 2.4, we discuss a non-zero-sum one-shot game in which player 2 is risk averse. In this setting, we show that the value of the game is positive under more relaxed conditions on the joint distribution of M and L. We conjecture that in a repeated game, player 2 cannot guarantee the value of the game to be zero by slightly modifying his optimal strategy in each stage. Given the complexity of analysing the repeated game and characterizing the price dynamics in this setting, we leave such work to further research.
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A partial characterization of the core in Bertrand oligopoly TU-games with transferable technologies

A partial characterization of the core in Bertrand oligopoly TU-games with transferable technologies

This result implies that outsiders’ strategy profile p N\S that best punishes coalition S as a first mover (α-approach) also best punishes S as a second mover (β-approach). 4 Partial characterization of the core In this section, we first provide an example in which the convexity does not hold for a large class of Bertrand oligopoly TU-games with transferable technologies. Then, we show that the convexity property is satisfied if firms’ marginal costs are not too heteroge- neous. Finally, even if the core can not be always fully characterized, we identify a subset of payoff vectors with a symmetric geometric structure easy to compute.
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An Augmented Lagrangian Numerical approach to solving Mean-Fields Games

An Augmented Lagrangian Numerical approach to solving Mean-Fields Games

Abstract: We present the application of the ALG2 algorithm [1] to solve numerically variationals mean field games [5]. Key-words: Mean field games, Augmented Lagrangian, Douglas Rachford Ce travail de recherche a reçu le soutien financier de l’ANR Isotace (ANR-12-MONU-0013). We acknowledge the financial support of ANR Isotace (ANR-12-MONU-0013) for this research. ∗ INRIA BP. 105 78153 Le Chesnay Cedex

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