Beyond Analytical Modeling, Gathering Data to Predict Real Agents' Strategic Interaction

Abstract

This paper presents research proposals on the interdisciplinary research infrastructure for understanding human reasoning in game-theoretic terms. Strategic reasoning impacts human decision making in social, economical and competitive interactions. The provided introduction summarizes concepts from AI, game theory and psychology. First result is a concept of interdisciplinary game description language as a part of the focused interdisciplinary research infrastructure. The need of this domain-speci c language is motivated and is aimed to accelerate the current developments. As second result, the paper provides a summary of ongoing research and its signi cance.

Introduction

Di erent scienti c disciplines predict outcomes of human social, economical and competitive interactions on the granularity level of individual decisions [1, p.4]. Autonomous intelligent systems, which perceive, decide and act in an environment according to their preferences, are called agents in AI, MAS and sociology. People and implemented arti cial agents are called as real agents.

A rational agent makes always decisions, whose execution has according to his subjective estimation the most preferred consequences for him [ 2,3 ]. Level of intelligence impacts the quality of the subjective estimation. Rationality justi es agents' decisions and predictions of other agents' decisions. In strategic interactions, agents are rational and apply mutually and even recursively the concept rationality. Game theory predicts outcomes of strategic interactions [ 3 ]. Further, game is a notion [ 4 ] for the formal structure of a concrete strategic interaction. Agents involved in a game are called players.

A game in normal form consists of strategies (sets of decisions) and players' preferences over outcomes [ 3 ]. Finite normal form games contain a nite number of outcomes. Game theory is there to provide equilibria id est to solve games. An equilibrium is an irrevocable combination of players' strategies { none of the players can improve his outcome by altering his chosen strategy. If players' preferences are de ned by their payo functions, the equilibria of nite normal form games are guaranteed [ 5 ]. A payo or also utility function assigns a numeric value as magnitude of preference to every outcome. Calculation of equilibria is generally NP-hard [ 6 ]. GAMBIT is a software library for this task [ 7 ].

In the case of players, who are intelligent enough to solve their game, an equilibrium should (immediately) occur independently of players' builds or even real identities. This is fundamentally di erent to the common AI approach (referenced further as game playing), where programs compete the last decades in playing chess e.g. still without achieving any irrevocable solution [2, chap.5]. Although the existence of an equilibrium is proven for chess, analytical abilities of none of the present game theorists su ces to show at least, whether the equilibrium implicates the win of the whites. If at least one of the players is not able to solve the game, an equilibrium is not guaranteed to occur.

Although the modern game-theoretic textbooks contain mostly ctive scenarios, the reader is asked to resist the conclusion that real-life cases can not be formalized as games. Contrary to parlor games, a concrete strategic interaction with lacking explicit rules is to be formalized as a game. Manual formalization is not guaranteed to be accurate and mostly results in far too simpli ed \toy games" as criticized in AI literature [ 8 ]. Scaling down the space of games can enable automatic formalization [ 9 ]. A player faces this problem as well as a strict outsider like a game theorist. He has to answer the question correctly, which interacting agents really exist in the environment and in what kind of game he is involved in. It is called incomplete information in game-theoretic terms [ 3 ], if this question can not be clari ed.

Incomplete information transforms to imperfect information, if a probability distribution over all possible formalizations of a game can be provided [ 10 ]. Poker e.g. is a game of imperfect information, where every player is supposed to be aware about the probability distribution over the possible hands, but does not know the current hands. It is not obligatory that a game becomes a common knowledge (CK) among its players { it is also possible that a player bears his own variant of the game in mind, he misconceives to be involved in. As a reminder, CK is that, what everybody knows and everybody knows that everybody knows it and in nitely prepending \everybody knows that" [ 3 ].

CK of the game would not exist during a poker round, if Alice hides cards in her sleeve unknown to her opponent Bob. Alice considers that Bob's actions should conform the equilibria calculated in the original game and she calculates the equilibria of a global game consisting of the original game and the \cheated" game [ 11 ]. Players may have an intractable number of mutual nested believes about the details of their game. For instance, Alice could reckon to some extent that Bob also cheats, Bob could suspect Alice of considering him as a cheater and so on. An intractable number of mutual nested believes results in an intractable size of the global game, where no game-theoretic solution is guaranteed to be calculated or even to exist in the case of in nity. Therefore, cases are concentrated on, where a game of imperfect information is its players' CK.

This paper concentrates on the relevance of strategic reasoning in real agents' decision making, which causes an (immediately) occurring equilibrium, if the unbounded intelligence is assumed. In the cases of absent strategic reasoning, game theory can still predict the direction of convergence id est towards an equilibrium. This is proposed by Price [ 12 ] for prediction of stochastic processes, more precisely populations in biological terms, and is called evolutionary game theory [ 13 ]. A convergence is also observed in the case of using reinforcement learning instead of strategic reasoning [14, e.g.]. The assumption of this paper is that the usage of the concept strategic reasoning can be extended by newer methods. This assumption does not negate the existence of reinforcement learning e.g. in real agents' strategic interactions and is discussed next in the case of people.

Although the real human preferences are a subject of philosophical discussions [ 15 ], the application of strategic reasoning assumes that they can be captured in concrete interactions as required for modeling rationality. The consideration of people as rational agents is disputed at least in psychology [16, pp.527{530], where even a scienti cally accessible argumentation exposes the existence of stable and consistent human preferences as a myth [ 17 ]. Since the last six decades nevertheless, the common scienti c standards for econometric experiments are that subjects' preferences over outcomes can be insured by paying di ering amounts of money [ 18 ]. However, insuring preferences by money is criticized by the term homo economicus as well.

The ability of identifying other agents and of modeling their reasoning corresponds with the psychological term ToM (Theory of Mind) [ 19 ], which lacks almost only in the cases of autism. For application of strategic reasoning, subjects as well as researchers, who both are supposed to be non-autistic people, may be then able of modeling of others' strategic reasoning too. In Wason task at least, subjects' reasoning does not match the researchers' one though [ 20 ]. People may use no logic at all [ 21 ], but also mistake seriously in the calculus of probabilities [ 22 ].

The data of econometric experiments does not match the equilibria of games according to which they are conducted [ 23,24 ]. That means that the strategic reasoning according to the global (researchers' point of view) game does not arise among the subjects due to a set of reasons, which should be clari ed. There is a need for more than only the pure analytical game theory, because even people familiar with game theory are observed to deviate from equilibria in multiple cases [ 24 ]. It is gathering and analyzing of data from experiments and eld studies. This paper steps further { it proves the potential of making the interdisciplinary research on real agents' strategic interactions more e cient. Like in bio-informatics [ 25 ], it is supposed to be done by an interdisciplinary research infrastructure { domain speci c languages and common tools.

The paper is organized as follow. Next section summarizes the main concepts for the interdisciplinary research infrastructure. Then, the section 3 presents detailed the hitherto research. At the end, the results are concluded in order to gure out the remaining construction sites. 2

Conceptualizing Interdisciplinary Research

Infrastructure

A conceptualization of the already partially existing (interdisciplinary) research infrastructure for real agents' strategic interactions follows. It aims to provide an elaborate overview and an exhaustive motivation. Arti cial agents are also included into consideration, although this paper concentrates mostly on people. People can be replaced by arti cial agents in order to simulate or to intervene human strategic interactions. Whether simulation or intervention { in both cases, arti cial agents can cut costs, allow a direct control of their builds, can be numerously deployed and are almost unlimited in period of use.

In order to avoid incomplete information, which can make the application of strategic reasoning intractable, an explicit formalization is to be used. Because a formalization of a concrete strategic interaction can be inaccurate, it is reasonable to execute it inversely. A concrete strategic interaction has to be created out of an already existing game. This is game realization and a software-based game realization is a game implementation [1, p.108]. Game realization is the almost always unmentioned action after mechanism design [2, p.632]. Mechanism design is inverse to game solving { adjusting of a game to already predetermined desired equilibria. If game realization is impossible, mechanism design is futile.

In real-life cases, games can be realized by physical conditions, by nonparticipating agents, by participating agents themselves or by a subset of the 3 previous instances [ 26 ]. If participating agents are responsible for a (partial) game realization, they should prefer the compliance with rules over the advantages from \cheating". A non-participating agent responsible for game realization can be modeled as rational too. For instance, the attorney in prisoner's dilemma keeps his word in order to not risk his reputation, where prisoner's dilemma is supposed to be familiar to the reader.

In the case of game implementation, the software can be divided into fractions: game management, game-solving algorithms, game-playing algorithms and auxiliary algorithms. Game management is the part of software, which executes the rules and calculates outcomes [ 27 ]. It may also record in order to gather data [ 28 ]. Human-computer interfaces aka proxy agents [ 29 ] are examples of auxiliary algorithms. Additionally, game implementation minus game-solving/playing algorithms is called game infrastructure [1, p.53].

In the case of explicit rules, the question about the form of games arises. The normal form is one of the two most general forms to express games in nonco-operative game theory. In co-operative game theory or also coalitional game theory [ 30 ], players may covenant and group into coalitions. The process of negotiating and the way of ensuring the agreements themselves are not issues of co-operative game theory. From non-co-operative game-theoretic point of view, disadvantages should arise for the players, who break the agreements. From both points of view, a player makes a rational decision, whether it is his own behavior or an agreement about a co-operative behavior. Both points of view are considered to be equal [ 3 ]. Due to the fact that the pure analytical game theory does not su ce, the interest is focused a more detailed form { the extensive form { the second most general to express games in non-co-operative game theory. In contrast to others, the extensive form captures separately the actions' sequences and their alternating subsequences in a representation known as game tree in AI. Therefore, the actions' sequences can be called as root paths. The normal form and the coalitional formalization are skipped as argued because of their higher abstraction. Also skipped is the consideration of continuous games [ 31 ], where equilibra can be acquired by solving of di erential equations and no general form exit.

The theoretical appropriateness of the extensive form does not result a computational one { the problem is the inappropriate size of a games represented in extensive form. For instance, one can consider that the rules of chess are sent as a game tree via network. It is rather an Interdisciplinary Game Description Language (IGDL) having expressive power of the extensive form, which is needed for 4 rough categories of considered instances within and beyond game implementations. These categories are A) game-solving algorithms, B) game-playing algorithms, C) non-solving/playing algorithms (game management, mechanism design, auxiliaries etc.) and D) scienti c human users (sociologists e.g.).

Less important issues are skipped out of this consideration for IGDL { issues like adjustments for evolutionary mechanism design [ 32 ] and for non-scienti c human users. Reducing the size compared with an equivalent representation in extensive form is called further compactness [1, p.65]. Compactness is not the only criterion for IGDL. For the categories A{D, one can summarize the partial criteria as following: 1. Computational speed-up. Regularities like symmetries can be used in order to reduce the computation time of equilibria [ 33 ]. This feature is captured by formalisms called succinct games and should be also provided by IGDL. Reduction of computation time by using compact representation applies also to game-playing algorithms [34, e.g.] and can be considered in the case of game management. 2. Re-usability & comparability. A language for games forces the re-usability and also comparability of game-playing algorithms as suggested by Pell [ 35 ]. This applies also for game-solving algorithms [ 7 ] and for non-solving/playing algorithms algorithms [ 36 ]. 3. Interdisciplinary human usability. IGDL should prevent the scienti c manual game-theoretic formalization from resulting in \toy games" as criticized in AI literature [ 8 ]. A graphical representation of a game may improve the usability even more [ 37 ]. 4. Decidability. This feature should be provided for IGDL in order to ensure that the calculation of outcomes de nitely terminates [ 38 ]. This also important for game-solving/playing algorithms to be able to calculate the consequences of actions. 5. General compact interchange format. For interfacing instances of the categories A{D, IGDL should satisfy the need of a compact interchange format. At same time, IGDL should be as general as possible, where the facility to express n-person games of imperfect information is most general. Finally, instances of all the mentioned categories should be at least theoretically IGDL-compatible in order to skip reformatting, id est to facilitate their e cient mutual integration. 6. Time. Time remains the issue disregarded by the extensive form. As one can conceive by comparing game playing in fast chess and in normal chess, time given for making decisions impacts them. Therefore, time is needed to be included in IGDL in order to ascertain the time dependent details by the explicit rules. Otherwise, the durations of actions' sequences e.g. may depend on the current game implementation and not be given explicitly in advance conjoined with the game.

Some examples for usage of IGDL can be provided. As 1st example, IGDLcompatible chess playing algorithms can be incorporated into system, which compete in playing other chess-like games described in IGDL. As 2nd example, a IGDL-based game editor can be used to allow non-computer scientists to set-up their own experiments. As 3rd example, data gathered in experiments conducted according to a game described in IGDL can be compared with the equilibria calculated by IGDL-based game-solving algorithms for the same game. As 4th example, game described in IGDL can be easily forwarded to the IGDL-based game-playing algorithms for an approximative solution, if IGDL-based gamesolving algorithms fail to output timely. As 5th example (proposed by [ 39 ]), the data of the experiments conducted based on IGDL can be better compared or even stored in a central web database like the state of art in bio-informatics. Games described in a special language with and without conjoined time dependent details are called further game descriptions. 3

Summarizing Ongoing Research

IGDL is the desired domain-speci c language, which has already some precursors and these precursors are summarized in this section. It is also possible that the ongoing research will bear di erent concurrent versions of IGDL. In order to assess the hitherto approaches better, a rough categorization of the used formal means is provided. Such categories are functions, graphs, logic, Petri-nets and so on. Tab.1 contains the regarded approaches and their rough categories. The ability to describe simultaneous moves (id est actions) is subset to more general imperfect information, because other players' actions can be unobserved only during a simultaneous execution. In deterministic games, it is impossible to describe a probability distribution over possible subsequences of actions.

In discrete non-co-operative game theory, there are di erent approaches for compact game forms, whose aim is computational speed-up [ 33 ]. The most important of them chronologically ordered are congestion games [ 40 ], sequential form [ 8 ], graph games [ 41 ], local e ect games [ 42 ] and action graph games [ 43 ]. The software GAMUT can generate random games of these and other kinds, where the extensive form is included [ 52 ]. Computational speed-up of sequential form in solving 2-person-games of imperfect information is used in GAMBIT [ 7 ].

GAme LAnguage (Gala) [ 44 ] is developed in order to provide an interface to the game-solving algorithms. Factually, Gala a ords Prolog-based game descriptions, where a game of extensive form or of normal form can be generated. The generated game-theoretic representation can be forwarded to GAMBIT or other game-solving algorithms. The main improvement of Gala compared to these representations is the game descriptions' compactness as perceived at least at the

Approach Crit. Means Class of games Citation Used congestion games 1,4 functions subset of n-person games, [ 40 ] A,C simultaneous moves sequential form 1,4 matrices 2-person games of [ 8 ] A,C imperfect information graph games 1,4 graphs, n-person games, [ 41 ] A,C functions simultaneous moves local e ect games 1,4 functions subset of n-person games, [ 42 ] A,C simultaneous moves action graph games 1,4 graphs, n-person games, [ 43 ] A,C functions simultaneous moves Gala 2 logic n-person games of [ 44 ] A imperfect information MAID 1,4 Bayes-nets n-person games of [ 45 ] A,B imperfect information continuous games 6 functions subset of 2-person games of [ 46 ] A imperfect information timed games 6 functions 2-person games [ 47,48 ] B GDL 1,2,4 logic deterministic n-person games, [ 38 ] B,C simultaneous moves GDL-II 1,2,4,5 logic n-person games of [ 49 ] B,C imperfect information game Petri-nets 1 Petri-nets deterministic n-person games, [ 50 ] A simultaneous moves PNSI 2,4{6 Petri-nets n-person games of [ 51 ] A{C imperfect information SIDL2.0 2,5,6 logic n-person games of [1, p.98] C imperfect information z-Tree-language 2,3,5,6 imperative n-person games of [ 36 ] C,D language imperfect information examples from Gala's software package. Due to full-scale Prolog, Gala does not provide decidability. Therefore, the generation of extensive form games from Gala game descriptions must not terminate.

There is a subset of game-playing algorithms, which is not only aimed to play games, but also to simulate human (strategic) reasoning in them. Simulating human reasoning falls in the subject of cognitive science. There are currently two di erent approaches, where the human strategic reasoning has to be expressed in a general language in order to facilitate an e cient comparability of the models. The rst is based on cognitive architectures [ 53 ][ 54 ], which are languages for models of general human reasoning. The second is based on Multi-Agent In uence Diagrams (MAID) [ 45 ]. The second factually conforms the game-theoretic point of view on strategic interactions and provides the alternative language MAID for describing games. MAID are shown to be expressive enough to represent n-person games of imperfect information in the algorithms for game playing. MAID can be also transformed to extensive form in order to solve them [ 55 ].

The previously discussed literature does not mention the inclusion of time dependent details in game descriptions, what some theoretical approaches from game theory, concurrency theory [ 56 ] and control engineering [ 48 ] aim. A current work [ 46 ] in game theory extends extensive form games of perfect information and continuous time [ 57 ] to such of imperfect information { continuous games. In [ 57 ][ 46 ], only a subset of such games is regarded { the set of player's actions is always the same. Generally, a point of time is assigned to every action and time grows strictly over a sequence of actions.

Game Description Language (GDL) succeeded in sparking an international programing competition on general game playing [ 38 ]. A concrete game description in GDL is sent to an arti cial game player and never to its human programmer { the programmer knows only the structure of GDL. This satis es the criterion 2. For the criterion 5, generality of IGDL's precursors is a trend { either it will be possible to describe n-person games of imperfect information in IGDL or IGDL should be extended to facilitate that. This trend caused the development of GDL-II [ 49 ], which is an extension of GDL for n-person games of imperfect information. Like GDL, GDL-II is based on Datalog, which is a version of Prolog [ 58 ]. Datalog guarantees decidability by banning functions, limiting variables' ranges and restricting recursion. Due to the decidability, the existing game management algorithm based on GDL-II is guaranteed to terminate. However, the ban of functions worsen compactness. For instance, if arithmetic addition is needed to describe actions' consequences in a game, the result for every required summands' combination must be separately de ned in the game description.

There are no time dependent details included explicitly in GDL-II game descriptions. GDL and GDL-II describe games in a way STRIPS-like [ 59 ] languages do. In languages for planning tasks, STRIPS-like descriptions can be replaced by descriptions based on Petri nets [ 60 ]. Petri Nets for Strategic Interaction (PNSI) is a game description language, which is proposed chronologically between GDL and GDL-II [ 37 ]. PNSI uses basic Petri nets instead of logic. Petri nets are also known being used in game theory to describe subclasses of games [ 50 ]. PNSI provides decidability [1, p.89]. The advantages of PNSI compared with GDL-II are the graphical representation of Petri nets and the ability to describe games of equidistant time. Equidistant time means that the game management algorithm for PNSI pauses exactly for one chronon between two game states [ 51 ], where a game state is also a node of its game tree. In this context, a chronon is a constant period of time, which is explicitly known to players. During a chronon, players' actions can be submitted. The game management algorithm for GDL-II is not of equidistant time, because the next state is calculated exactly after the submission of the last action, if it is inside the allowed time period. The time point of the last submission may vary depending on players. Of cause, GDL-II is supposed be also able to describe games of equidistant time, if its game management algorithm is modi ed as proposed for PNSI.

For PNSI, there exist an algorithm that can generate games of extensive form from game descriptions as in the case of Gala [ 51 ]. Therefore, PNSI provides an interface to game solving algorithms. A game of extensive form generated based on a PNSI game description is a slightly modi ed state transition system of the underlying basic Petri net. In this context, a state transition system is an oriented graph consisting of game states, where every edge represents a progression in time. There is still no algorithm for GDL-II to generate games of extensive form. A state transition system can be also generated for GDL-II game descriptions [ 49 ]. Therefore, generation of extensive form games is supposed to be also possible for GDL-II game description.

PNSI su ers of insu cient compactness as well as GDL-II but in a di erent way. The arithmetic addition and subtraction are banned in Datalog, but inherently supported by basic Petri nets. On the other hand, basic Petri nets require every state to be decoded as a vector of natural numbers and do not have other operations than the addition and the subtraction. The game description of the parlor game Nim needs in PNSI much less space than in GDL-II [ 1 ]. The opposite for chess, a GDL-II chess description needs less.

Dropping the criterion of decidability may dramatically improve compactness. Strategic Interaction De nition Language (SIDL) is based on ISO-Prolog, does not provide decidability and attains for example games a higher compactness [1, p.98]. Of cause, the widely used game description in an imperative language can be also mentioned. However, the game management part of software aka game server is then required to send its own code in order to provide explicit rules [1, p.54].

For scienti c human users beyond computer science, there is an ongoing development of user-friendly software for experiments [ 61 ]. RatImage [ 62 ] and TEEC [ 61 ] are examples of the rst generation of such software. They are libraries, which facilitate programming. The second generation provides already domain-speci c languages for the game management and the layout of humancomputer interfaces. These are ComLabGames [ 63 ] and z-Tree [ 36 ]. z-Tree is the most used [ 39 ]. There is a z-Tree-language, which is actually an imperative language, in which the game and also the human-computer interfaces can be described. This language does not provide decidability. It has no relations to game solving or playing algorithms. 4

Conclusion

A large-scale view on the problem of understanding human strategic reasoning is presented. The elaborated solution is the development of the interdisciplinary research infrastructure. This research infrastructure is proposed to make the interdisciplinary research more e cient, as it is already observed in similar interdisciplinary problems. As an underline of the large-scale view, there are matters chained from di erent sources, which have been never cited together before. For instance, GDL-II and z-Tree are such matters.

The elaborated concept is the domain-speci c language IGDL. A summary of its precursors shows that none of these is developed enough to satisfy IGDL's full outline. There is still no language, which incorporates the graphical representation like PNSI, compactness improvements like GDL-II and a proven human usability like z-Tree-language.

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