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1 Avalable onlne at ScenceDrect IFACPapersOnLne 49 (06) 09 0 Addng Informatonal Belefs to the Players Strategc Thnkng Model 06, IFAC (Internatonal Federaton of Automatc Control) Hostng by Elsever Ltd. All rghts reserved. Currently these models are beng actvely developed n behavoral game theory as a tool to predct human s behavor. These models have many applcatons n such areas as markets (Wrght et al. 0), barganng (Wrght et al. 0), auctons (Crawford & Irberr 007), lotteres (Östlng et al. 0), and as a tool for control problems (Novkov & Korepanov 0). On the other hand, they are beng mplemented n multagent systems to specfy the behavor of software agents whch act n place of human decsonmakers (Wunder 0, Sonu 05). The nformaton reflexon model (Novkov & Chkhartshvl 04a) s used for games wth external parameters on whch players utlty functons depend. Players are nterested to fnd the values of these parameters and what opponents know about the parameters and what opponents thnk about ther opponent s knowledge of the parameters etc. Hstorcally, models of strategc and nformaton reflexon have been studed ndependently. In ths paper we propose a general model whch allows descrbe both types of reflexon smultaneously.. Strategc reflexon model. MODEL Here we descrbe the basc and well known strategc reflexon model. Alexander G. Chkhartshvl*, Vsevolod O. Korepanov* * V. A. Trapeznkov Insttute of Control Scences of Russan Academy of Scences, Moscow, Russa (emal: sandro_ch@mal.ru, vkorepanov@pu.ru). Ths work s partally supported by the Russan Scentfc Foundaton, project no Abstract: We propose the model that connects the strategc reflexon/strategc thnkng model and the nformaton reflexon model of players n game theory. Ths model can descrbe both types of belefs: players belefs about opponents thnkng models and players belefs about ther opponents awareness about the external parameter value. We consder some general examples and also the example of the Cournot olgopoly model. Keywords: cogntve control; smart nfrastructure; strategc reflexon; strategc thnkng; nformaton reflexon. For smplcty, consder a game of three players n the normal. INTRODUCTION form G = (N, A, u), where N = {,, } s the set of players, A N A s the set of possble acton profles wth A In game theory there exst several types of thnkng models of beng the set of actons avalable to player, and u { u } N players wth bounded ratonalty. Among them there are the s the set of utlty functons u strategc reflexon/strategc thnkng model and the : A R mappng the acton nformaton reflecton model. The basc strategc profle to the utlty for player. Players act smultaneously, reflexon/strategc thnkng model could be modfed to ndependently and oneshot by choosng an acton profle mprove ts predctve power (Novkov & Chkhartshvl from ther acton sets. The result of the game s the acton 04b). profle a=(a,a,a ) of actons of three players. We fx the acton profle a 0 = (a 0,a 0,a 0 ) A whch corresponds to an obvous (n some sense) acton profle and whch may reflect such qualtes of a player as farness, dscreton etc. (we only consder games that have at least one such an obvous profle). Defnton. Defne the best response functon of player br : A A : br ( ) max (, ) a C Arg u y a, y where a A, a s the jont actons of all players except for and C s a choce functon that unquely pcks one acton from a subset of actons. We say that the player has the zero rank ( rank 0 ) of reflexve thnkng f he chooses the acton a 0. Some player may have a belef that all hs opponents have rank 0. Then he predcts (perhaps, erroneously) opponents actons, and hs subjectve optmal acton s: a br ( a 0 ), () where a s the acton of the player, superscrpt means that we say that such player has rank. If there s two players and j of rank, then they have wrong belefs about each other: thnk that j has 0 rank, and j thnk that has 0 rank. In other words we can say that there s the real player j and the phantom player j exstng only n the mnd of player. The same true for player , IFAC (Internatonal Federaton of Automatc Control) Hostng by Elsever Ltd. All rghts reserved. Peer revew under responsblty of Internatonal Federaton of Automatc Control. 0.06/j.facol.06..8
2 0 Alexander G. Chkhartshvl et al. / IFACPapersOnLne 49 (06) 09 0 Belefs of the player can be represented as an orented tree whose nodes are players, real or phantom (.e. exstng only n the mnd of other players). An arrow from player A to player B would mean that player B knows all nformaton about player A (herenafter, we wll call a real or phantom player just a player, unless the opposte s stated). Such a tree, as an example for =, s shown on Fg.. So players of rank and more must have belefs about belefs of ther opponents and so on (these are players, etc. on Fg. ). For a complete descrpton of the stuaton we need to defne belefs of all players whch creates a herarchc nformaton structure. An example of such a structure s gven n Fg.. Fg.. An example of a belef tree. On the dagram, node s a real player, nodes ( onetwo, not twelve) and are players and n belefs of player. Note that nodes and are hangng,.e. that phantom players and do not have belefs about opponents. Ths fact means that such players have rank 0 n our model. It s generally assumed (Novkov, Chkhartshvl 04b) that the rank of a player s one greater than the largest rank that hs opponents have, n hs opnon. So player on Fg. has rank. If player thnks that hs opponents have rank, for example, then he has rank and hs subjectve optmal acton s a br ( a, a). () Smlarly, f player thnks that hs opponents and have ranks 0 and, then he also has rank and hs acton s descrbed as 0 a br ( a, a). () Graphs for these cases are shown on Fg.. Fg.. All players belefs example We are gong to merge some players as they have equal belefs and, consequently, equal actons. For example, and on Fg. are phantom players wth rank 0 who act dentcally, so we can merge them n one node. In other words, players and have equal belefs about player. Furthermore, f we dentfy n ths way all rank 0 players, we can see that players and have equal belefs about player. After completng all mergng operatons, we get a more vsual graph of players belefs, the graph of a reflexve game (Novkov Chkhartshvl 04a), where each node represents a real or a phantom player (Fg. 4). (a) Fg.. (a) belefs of player that hs opponents are rank players, (b) belefs of player that hs opponents has rank 0 and. Thus, the rank of a player equals 0, f he s a leaf node of a tree (there are no ncomng arrows), otherwse t equals to the number of arrows n maxmal length path from termnal nodes to the player. (b) Fg. 4. The graph of a reflexve game. As we can see, the graph of a reflexve game dsplays the followng nformaton: ranks of players; adequate or nadeqate belefs: player s adequately nformed about player (.e. the mage of player n mnd of player matches wth real player ), but he s not adequately nformed about the player.
3 Alexander G. Chkhartshvl et al. / IFACPapersOnLne 49 (06) 09 0 equvalent belefs: players and have the same belefs about the player (though ncorrect). Note that doublesded arrows are not possble n our model unlke n an arbtrary the graph of a reflexve game. Ths follows from the assumpton that each player thnks that hs opponents have a lower rank and rank 0 players do not have belefs about opponents at all.. Informatonal reflexon model To develop our model further, we append the nformaton reflexon to the exstng herarchc belefs structure. Let θ Θ be the external system parameter on whch players utlty functons are dependent on: u : Θ A R. The second assumpton s that players have common knowledge that the parameter value s not known exactly each player can estmate t dfferently. In other words player have belef about θ: θ Θ. Now, to predct player j best response acton a player must have an opnon about opponent s belefs about the value of θ: θ j Θ and hs belefs about hs opponents belefs and so on. We can put smple nformaton reflexon model to our example case (fg. 4). Let all players beleve that θ = θ a Θ. Then t can be dsplayed as the same graph as n fg. 4 equal belefs of all are not shown by conventon. Here θ = θ = θ = θ = θ = θ = θ = θ = θ = θ a. Let all players beleve that θ = θ a but player beleves that θ = θ b θ a. Now we cannot merge players and snce they have dfferent belefs (equal strategc but dfferent nformaton reflexon), and so they wll be represented by dfferent nodes n the graph of a reflexve game (opposte to the stuaton n Fg. 4). Ths s shown on Fg. 5 (here, belefs of most players θ = θ a are not shown). Now players and have dfferent belefs about player. Despte the fact that player thnks that θ = θ b, he beleves that hs opponents (players and ) thnk that θ = θ a. of the nformaton reflexon model and the strategc reflexon model.. EXAMPLES Consder the classc Cournot olgopoly model for three players, a sngleproduct market wth three manufacturers (the players ). Each player can manufacture the product n the total amount ( volume ) a > 0. Let market prce be defned by the expresson N M b a, (4) where M s the market capacty and b s the prce elastcty. When producton costs are not taken nto consderaton, the player s utlty can be expressed as u a, a, a ) a ( M b a ). (5) ( N. Example. Strategc reflexon We agan start wth ntroducng the strategc reflexon. Snce all players are n equal condtons, we can assume that a rank 0 player thnks all players would act the same way and wth maxmum utlty for each. Therefore the acton profle of rank 0 players s a 0 = (a, a, a) where a s an acton that maxmze utlty u (a,a,a)= a(m ba) for all. So a 0 arg max u ( x, x, x) arg max x( M bx) M /(6b) x0 x0 Let the players belefs be defned by the graph n Fg. 6. (6) Fg. 6. The graph for the Cournot olgopoly model n Example. (θ b) Fg. 5. The graph of a reflexve game wth nformaton reflexon. It s clear that ths way we can specfy an arbtrary belefs structure about the parameter θ,.e. such a structure n whch all players have defnte belefs about θ and about the opponents belefs about θ and so on, see Novkov, Chkhartshvl (04a). The resultng model s a combnaton We can calculate players actons movng from the bottom to the top n the gven graph of a reflexve game; the results are shown n Table. To calculate expected payoff of player we take hm acton a = br (a,a ) (hs best response to hs belefs) and actons of players and (actons that expect from opponents) and put them to utlty functon u (a,a,a ). Real payoff s calculated only to real players, by put real actons to utlty functons: u (a,a,a ). player acton Table. Example expected payoff real payoff a 0 = M / 6b M / (b)
4 Alexander G. Chkhartshvl et al. / IFACPapersOnLne 49 (06) 09 0 a 0 = M / 6b M / (b) (5Mm) / b (5Mm) / ( b) a 0 = M / 6b M / (b) M / (8b) br (a, a ) = M / b br (a,a ) = M / b br (a,a ) = M / 6b M / (9b) M / (9b) M / (6b) M / (9b) M / (8b) It s nterestng that actons of players and turned out to be equal. Thus, player appears as f he s adequately nformed about player, and consequently he wns more than others (player s adequately nformed about player ). Moreover, expectatons of player about hs payoff and opponents actons (f t s allows by the game setup) wll match realty.. Example. Strategc and nformaton reflexon Now we can ntroduce the nformaton reflexon. Assume that the value of the parameter M s not known to players and each player has a belef M about t. Assume M {M, m}, and n realty the market capacty s M. It s possble that the awareness s as gven n the table n Fg. 7 (here only wrong belefs about the market capacty are marked (m) ). (m) (m) Fg. 7. The graph for the Cournot olgopoly model n example. Table. Example (m) (m) player acton expected proft real proft a 0 = M / 6b M / (b) a 0 = M / 6b M / (b) a 0 (m) = m / 6b m / (b) a 0 (m) = m / 6b m / (b) (Mm) / 6b (Mm) / (6b) (mm) / 6b (mm) (5M m) / (6b) M / 4b M / (6b) M(4M9m) / (4*48b) (Mm) / 4b (Mm) / (6b) (7Mm) / 4b (m9m) / 48b (7Mm) / (4 b) (m9m) / (48 b) (7Mm) (4M 9m) / (4*48b) (m9m) (4M9m) / (48 b) As seen on Table, ths stuaton s more complex and n general all players expectatons wll not concde wth realty. It s clear that belefs of each player can be set arbtrarly and that the expected and real payoff can be calculated accordng to those belefs. 4. CONCLUSIONS The general model combnng strategc and nformaton reflexon s presented. It turns out that such a combnaton does not add more complexty; t may only ncrease the number of nodes of the graph of a reflexve game. We expect that the deas and results from the models of both reflexve types wll extend to ths new model. Ths may nclude, for example, the common knowledge of two players (whch can be represented as a double arrow n the graph of a reflexve game) or unfed model of all players of the same rank (such as n levelk, cogntve herarches, and reflexve parttons models). (Novkov Chkhartshvl 04b; Wrght et al. 0). The advantage of the model s that t descrbes a broader class of strategc belefs that can be modeled n one framework. We hope that ths model could be used as a tool for settng up, analyzng and vsualzng agents strategc belefs n multagent systems and n game theory. REFERENCES Crawford, V., and Irberr, N. (007). Levelk auctons: Can boundedly ratonal strategc thnkng explan the wnner's curse and overbddng n prvatevalue auctons? Econometrca, 75, Novkov, D.A., and Chkhartshvl, A.G. (04a). Reflexon and control: mathematcal models. CRC Press/Balkema, Leden. Novkov, D.A., and Chkhartshvl, A.G. (04b) Mathematcal Models of Informatonal and Strategc Reflexon: a Survey. Advances n Systems Scence and Applcatons,, Novkov, D.A., and Korepanov, V.O. (0). The reflexve parttons method n models of collectve behavor and control. Automaton and Remote Control, 7 (8), Östlng, R., Wang, J.T., Chou, E.Y., and Camerer, C.F. (0). Testng Game Theory n the Feld: Swedsh
5 Alexander G. Chkhartshvl et al. / IFACPapersOnLne 49 (06) 09 0 LUPI Lottery Games. Amercan Economc Journal: Mcroeconomcs, (),. Sonu, E., Dosh, P. (05). Scalable solutons of nteractve POMDPs usng generalzed and bounded polcy teraton. Autonomous Agents and MultAgent Systems, 9(), Wrght J.R., and LeytonBrown K. (0). Behavoral gametheoretc models: A Bayesan framework for parameter analyss. Proceedngs of the th Internatonal Conference on Autonomous Agents and Multagent Systems,, Wunder, M., Kasers, M., Yaros, J. R., & Lttman, M. (0). Usng terated reasonng to predct opponent strateges. In Proceedngs of the 0th Internatonal Conference on Autonomous Agents and Multagent Systems,,
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