 Applied Mathematics, 2011, 2, 1182-1190 doi:10.4236/am.2011.29164 Published Online September 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Multi-Team Bertrand Game with Heterogeneous Players Mohammed Fathy Elettreby1*, Daoud Suleiman Mashat2, Ashraf Mobarez Zenkour2 1Mathematics Department, Facu lty of Science, King Khaled University, Abha, Saudi Arabia 2Department of Mat hem at i cs, Faculty of Science , King Abdulaziz University, Jeddah, Saudi Arabia E-mail: *mohfathy@mans.edu.eg Received March 9, 2011; revised June 8, 2011; accepted June 15, 2011 Abstract In this paper, we proposed a general form of a multi-team Bertrand game. Then, we studied a two-team Ber-trand game, each team consists of two firms, with heterogeneous strategies among teams and homogeneous strategies among players. We find the equilibrium solutions and the conditions of their local stability. Nume- rical simulations were used to illustrate the complex behaviour of the proposed model, such as period dou-bling bifurcation and chaos. Finally, we used the feedback control method to control the model. Keywords: Bertrand Game, Non-Convex Dynamical Multi-Team Game, Incomplete Information Dynamical System, Marginal Profit Method, Nash Equilibrium 1. Introduction Game theory [1,2] is the study of multi-person decision problem. Such problems arise in economics. The game is called incomplete information if at least one of the players does not know the other player's payoff, such as in an auction when the bidders do not know the offers of each other. Otherwise, it is called complete information game. Also, the game can be classified to static or dyna- mic game. There are two famous economic games, the first is the Cournot game  and the second is the Bertrand game . In economic games, the first step is to construct the game. The second step is to solve the game (get their Nash equilibrium) and study the stability of these equilibria. Nash  showed that in any finite game there exists at least one Nash equilibrium. Nature push us to make teams in all fields. This has at least two main advantages. The first is the improvement of our profit and the second is that living in a team reduces the risk. For example, in the forest animals live in teams (herds). Since looking for food in a team is more efficient than doing it alone and reduces predation risk due to early spotting of predators and that existing in a team gives a higher probability that the predator will attack another member of the team. Another example is the competition between firms in the market. Suppose M branches of McDonald fast food shops compete against branches of Kentucky fast food shops. NMulti-team game has been studied in . In their work, they proposed and applied the concept of multi-team game in the hock-dove game, prisoner dilemma game and Cournot game. Also, the Cournot multi-team game has been studied in [7-11]. The standard static Bertrand game has been studied in . A duopoly Bertrand game with bounded rationality is studied in . Multi-team bertrand game is studied in  with two teams but the second team consists of one player. We will construct the model in Section 2. In Section 3, we will analysis the model, i.e., we will find its equi- librium points and their stability conditions. Some nume- rical analysis will be done in Section 4 to show the com- plexity behaviour of the model. Finally in section 5, we will use the feedback control method to control our odel. m 2. The Model Bertrand game is a model of competition used in eco- nomics. It describes interaction among firms that set prices and their customers that choose quantities at that price. In this game there are at least two firms producing homogeneous products and compete by setting prices simultaneously. Consumers buy everything from a firm with a lower price. If all firms have the same price, consumers randomly select among them. Suppose there are totally firms (produce certain product) in the market and these firms are divided into teams. Let ij be the price per unit of that product produced by the firm in the team and let ijc be the marginal cost of producing one unit of that product nNpj i M. F. ELETTREBY ET AL.1183  by the firm in the team . Then, the payoff of the firm in the team , if it played without the team, is given by the following equations; j =iji1,,aiji, pcπ=,1,, =,ij ijijlkijibpppiNjN (1) where is the number of teams and i is the number of firms in the team . The positive constants are the demand parameters where is the slop of the demand function. N N,ab bWe propose that, firms in the same team share some of their payoffs with their team mates. So, let ilj be the payoff rate that firm will takes from the payoff of firm in the same team . It is clear that , jilj=11212lilji0<<1ilj<1lof the first firm of the first team , if he played with the team, is given by the followimg; and . For example, the final payoff <1jj=1iπi11 1131 11111 121311 111=1 ππ,NNN  1113ππlj where 1 is the number of firms in the first team. In general, the final payoff of the firm in the team i is given by; Nj=1 πNNiiiijjlijlj illj  (2) In the case of two teams =2N=2 where each team consists of two firms and from Equation (1), we get the following payoffs of the firms in each team; 12=NN11===bbbb,,.1221221111 1112212212121211 21 222121212211 1222222221 11 12=,pca p p pppcapppppcapp pppcappppππππ (3) Using the assumption in Equation (2) of sharing some of the payoffs and Equation (3), we get the final payoffs of the firms as follows: 1111121121 121112211212 112221122121 222222212212 21=1 ππ,=1 ππ,=1 ππ,=1 ππ.     (4) In this model, we assume that the firms in the first team use the marginal profit method , to expect their profit for the next time according to the following equations; 1111111=, tjtt tjjjjtjpp pjp=1,2, (5) where 1j is the speed (rate) of adjustment and it is a function of the price 1jp. The firms in the second team use Nash equilibrium , to make their decision for the next step by solving the following equations; 22=0, =1,2.tjtjjp (6) In this model, we assume that the speed of adjustment will be linear and take the form 11 11=ttjjjppj, and 1>0j. Substituting from Equation (4) in Equations (5) and (6), we get the following system (7); Then, Equation (7) describe a system of two teams, each team consists of two firms with homogeneous strategies among each firms in each team and heteroge- eous strategies among teams. n 3. The Analysis of the Model The steady state (equilibrium) solutions are very interest . In the context of difference equations, an equilibrium solution x is defined to be the value that satisfies the relations 1==ttxxx. Then, we can get the equilibrium solutions for our model by the following. Let   11 1111212122tp111112111112212221 121211 1121221121211 21 2212111122122111222 22212=1 2,=1 2,1=,221tt tttttttt tttttttttttppabcbpppp pcppabcbpppp pccppp pcbb  111221ppbpa 21221 11 1221212211=.221ttttabpppp cbb122c (7)Copyright © 2011 SciRes. AM M. F. ELETTREBY ET AL. Copyright © 2011 SciRes. AM 1184 ,.1111 111112121211212121 222222==,====,==tt tttt ttpppppppppppp (8) Then, using Equation (8), the equilibrium points are given by solving the following system: 1111 11121111122122211212111212211212 11 21 22121111221212122111222 22212212222221 11 122122112121=,2211=221ttpabcbpppppcpabcbpppppcpabcppp pcbbpabcppppbb    21 .c=0,=0, (9) We get three boundary equilibrium solution points and the fourth one is the coexistence equilibrium one. The first boundary equilibrium one is given by 12111=0,0, ,,llEdd where 22 222 2211221211212=4 1111,db 21 222222222222121 121221211221121221222112=12112 11111,labc bcb  221  2222 2222222221221211221122112 222112212112=1211112 111.la bcbcb  2 The second boundary equilibrium one is given by 1221222=0,,,,nnEpdd where  2122 22222221122112212112=1411112 1212 12,db bbbb  222 11121 12212112112112222 2221 21122112122112222 22121 22211221=2 11211 21121411112 122114112 12,nbba bcbcbcbbb bbcbbbb      112   222 1121221122112112112222 221122112211212222 2221 221221212112=2 11211 2112114112 12 2141111212.nbbabcbcbcbbbbbcbbb b     112 and 112 21212 1112 1221=.221abc dnncpbd b The third boundary equilibrium one is given by 1231133=,0,,mmEpdd, where  2122 22222312122112212112=1411112121212db bbbb   222 11121 12211212112112222 2212 21122112122112222 21221 22211221=2 1121121121411112 122114112 12,mbba bbccbcbbb bbcbbbb      112 M. F. ELETTREBY ET AL.1185   222 112122112121211211222221212 21122112122222212 221221212112=2 112112(1)12114112 1221411112 12.mbba bbccbcbbbbbcbbb b      112 and 111 31221 1211 1312=.221abcdmmcpbd b The most important one is the coexistence one 4444411122122=,,,Epppp,where  43113443 12344321344322344341143 3421111 221121 22 121122 122122 141221 12431134431234432134 434212= ,2= ,2=aBBcDBFBcEBHBcFBDBcHBEBpAB ABaBBcDBFBcEBHBcFBDBcHBEBpAB ABaAAcDAFAcEAHAcFADAp     22 344343 3421111221121221 211221 22122141221 12,2= ,cHAEABA BAaAA cDAFAcEAHA cFADA cHAEApBA BA  where, 121 11111122112 2112 211111221 12321 11111122112 2112 211112112 21222 22222122112 2112 212221221 12=81 1211,1121=81 1211,(1) 121=81 1211,(1) 121AbbAbbBbbB      422 22222122112 2112 212222112 21=81 1211,1121bb    211 111112211212 21211111221 12=41 211211,1121Abb   411 111112211212 21211112112 21=41 211211,1121Abb     122 222212211212 21212221221 12322 222212211212 2121222211221=41 211211,1121=41 211211,1121BbbBbb     1112 2111121 1221111112211221 1231112112 212212 2122221 122222 2212211221 1242222112 2121=,12 (1)4112 1=,112121=,12 14112 1=,1121bDbbDbbDbbDb 1112 2131112 212111 1112212112 2111111221 1221=,12 14112 1=,1121bEbbEb Copyright © 2011 SciRes. AM M. F. ELETTREBY ET AL. 1186 2212 2142212 212222 2212212112 2122221221 1221=,12 14112 1=1121bEbbEb,   2212 2112221 1222 212 12213222112 211112 2121121 1211 112 12214112112 212221 1232212 21221 21121=,12 121 1=21112121=,12 121 1=,112121=,12 121=bFbbFbbFbbFbbHbH    , 22122221221 121121 1241112 2111 121 211221111221 121,112121=,12 121 1=.1121bbbHbbHb   The stability of this equilibrium solutions is based on the eigenvalues of the Jacobian matrix of the system (7), which is given by (10); where 1112 11111221121 1212=14 tt tttabcbp p pppc22 and 12211212112122112 11 11=14 tttttabcbpppppc . The equilibrium solution will be stable if the eigenvalues , =1,2,3,4ii of the Jacobian matrix (10) satisfy the conditions <1, =1,2,3,4ii. The eigenvalues for the first equilibrium point are given by 1E11111, 21212 , and 22 22122121123,4 22 212 211141 1b   . For the other equilibrium points it is very difficult to compute these eigenvalues. Instead, we find the characteristic polyno- mial, which has the following form: 432123=,Paaa  4a Then, the necessary and sufficient conditions , for all roots of the characteristic polynomial P to satisfy the conditions that <1, =1,2,3,4ii are the following: 1234123442434122243412224 3411431)1= 1>0,2) 1=1>0,3)< 1,4) 1>,5) 1>1PaaaaPaaaaaaaaaaaaaaa aaaaaa . (11) The coefficients of the characteristic polynomial for the most important one (coexistence) are as follows:  1111122211112211 1111 1211 12122121121111 111212122122 221221211222 212 21=2 ,=1 1111111,411aappppbbb      11 111111111 12211111 12 1111 12111 112 12211212212 122112 12212212 212122221 1222111111 11111 ,022 21111 02221pppppbb bbbb     1p (10)Copyright © 2011 SciRes. AM M. F. ELETTREBY ET AL.1187    122222211 1112122112211221322 212 211222 22212 122112211221122122 212 211111 1111 1211 12122112211221111 11122111 11=41 1111 1141 111111112pabpbppbp  1212122111111221221122122 212 211111211 ,21 1pbb  222  11122 2211 12 11 1221122121121221422221 12111 222211 12 11 121221122112122122221 1211 11221112111221 12122121 1211 11=41111 114111111ppabppbpp           2212 2122212 211 122 2222 2211 111212212122111121 12122121 1212212222221 1221 1222 2211 112221121221221241 111 1111 11411 41111 11411bppbbb 2            221. Then, the equilibrium solution 4 of the system (7) is stable under the conditions (11). This means that in the long run all firms are coexist. So, the market will be stable. E 4. Numerical Simulations In this section, we will use some numerical simulations to show the complicated behaviour of the model (sta- bility, period doubling bifurcation and chaos). Figure 1 shows the bifurcation diagram of the prices and profits with respect to the adjust speed 11 while the other parameters are constant and have taken the values , , , , , , , , , , , , , and 011 =0.30p112 =0.212 =0.13c12 =0.2012 =0.55p121 =0.3 21221 =0.21 c021p=0.1 =0=0.2=0.602213a022 =0.64p411 =0.1c=3bc.=1122. This figure shows that the equilibrium point = 0.5212497,0.5254234,0.5638708,0.5599204p is locally stable for 11 < 0.7569938, after this value it became periodic and finally the system became chaotic. The same thing occur to the profits in figure 1B at the same value of 11. Figure 2 shows the effect of changing the parameters ilj. We get a bifurcation diagram for the prices and profits with respect to 112 with the values of the other parameters are the same as in Figure 1 except that 11 became constant and takes the value 12 =0.9 and 112 became variable. We note that the small cooperation among the firms in the same team () will lead to a complex behaviour in the system, while the increasing this ooperation will lead to the stability. 112< 0.3459991c 5. Chaos Control As we seen in the last section, the adjustment rate ij and the payoff return ilj of the boundedly rational firms play an important role in the stability of the market. So, to avoid this complexity we will try to control the chaos. We will use the feedback method  to control the adjustment magnitude. Modifying the first equation in our system will give us the following controlled system; Copyright © 2011 SciRes. AM M. F. ELETTREBY ET AL. Copyright © 2011 SciRes. AM 1188 Figure 1. The bifurcation diagram of the prices and the profits with respect to α11. Figure 2. The bifurcation diagram of the prices and the profits with respect to 112.   11 11111111112111112212221 121211 112121212211212 11 21 221211112121212122111222 222122=1 21=1 21=,221tt tttttttt tttttttttttpp pabcbpppppckpp pabcbpppppcpabcppppcbbp ,,2112222 21111221212211=,221tttttabcpppp cbb  (12) where the parameter is the control factor. The Jacobian matrix of the controlled system will be: >0k M. F. ELETTREBY ET AL.1189  111111 11122111 111211 111211 1111 112 12211212212 12211212212212 212122221 1222111111111 1111022 21111 02221pppkkkkppbb bbbb    111p (13) The original system is chaotic for the parameter values , , , , , , , , 11, , 21 , 22 , , , 11011 =0.30p112 =0.212 =0.13c=1.1012 =0.55p121 =0.3 212=0.21c021 =0.60p=0.1 221 =0=0.23022 =0.64p4=0.1c=1 =3b.a1c and 12 =0.2. But the controlled system is stable (i<1, =1,2,3,4i) for all the above parameters values and for . > 0.455k9977 Figure 3. The bifurcation diagram of the prices with respect to the controlling factor k. Figure 4. The bifurcation diagram of the prices with respect to the controlling factor k = 0.5. From Figure 3, we find that the controlled system begin chaotic, periodic and then stable by increasing the control factor . kFigure 4 shows the stability behaviour of the controlled system when . 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