CC BY
2Hong-wei Gao1, Ye-ming Dai1 and Han Qiao2
1 College of Mathematics, Qingdao University, Qingdao,266071, P.R.China, E-mail: gaosai@public.qd.sd.cn 2 College of Economics, Qingdao University, Qingdao,266071,P.R.China
Abstract By introducing point (state) payoff vector to every point node on connected graph in this paper, dynamic game is researched on finite graph. The concept of strategy about games on graph defined by C.Berge is introduced to prove the existence theorem of absolute equilibrium about games on connected graph with point payoff vector. The complete algorithm and an example in three-dimensional connected mesh-like graph are given in this paper.
Keywords: connected graph, point payoff vector, simply strategy, absolute equilibrium, three-dimensional mesh-like graph.
1. Introduction
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The game tree which mainly describes the process of dynamic games is a kind of simple structure graph. Therefore, the research results of game on graph could generally popularize the category of dynamic game. The original type of games on graph, whose definition was given by C. Berge, is the games on finite tree (E.Zemelo, 1961). Literature (Rozen, 2005) discusses games on graph whose target structure is defined by coherent relation of terminal point set. The author extends C. Berge’s concept, namely, on every given point of graph, the choice for the next point is determined by the former experienced points, rather than only determined by the last point that the player had just reached. The results on literature (Berge and Ghouila-Houri, 1965; Rozen, 2005)are both given on the two- dimensional graph for games with terminal payoff. The point payoff vector is introduced to every point point on finite graph is expected in this paper. The absolute equilibrium of dynamic games is researched by applying the concept of strategy of games on graph defined by C. Berge. The related tasks finished by the author contain the following:
(1) By establishing the corresponding relations between situation and game tree on two- dimensional directed graph, games on directed graph is transformed to game tree. Also, the algorithm is given, and the Shapley vector is chosen as the cooperative solution of the two-dimensional directed graph.
(2) Partial cooperative dynamic games is studied on two-dimensional mesh-like finite graph. Players adopt partial cooperative behaviors rather than completed cooperative behaviors. The main feature of partial cooperation is that behaviors of each player are the combination of cooperative behaviors and individual behaviors.
* This work was supported by National Natural Science Foundation of China under grants No.70571040,70711120204 and 70871064.
Also, algorithm of the solution of partial cooperative games and the optimal path are given on two-dimensional directed graph.
Symbol system, strategy of plays and point payoff of games on graph are given on the second part of this paper. For the third, the existence theorem of absolute equilibrium about games on connected graph with point payoff vector is proved. On the fourth part, the complete algorithm of absolute equilibrium is constructed. For the last part, the author gives an example of absolute equilibrium about games on three-dimensional connected mesh-like graph with point payoff vector.
2. Symbol and Definition
Connected graph with point set A is written as < A, 7 >, where 7 C A2 ( 7 is the arc set of the connected graph) and point set is A = (a0, a1;... }. 7 < a > denotes all the point sets after point a, and 7' < a > is the immediate subsequent point set of point a. Af is written as the point set which has not subsequent points. < A, 7 > is called n point graph, if subdivision of A\Af is given, which is (A1,..., An}. Using the term of games, set of players is N = (1, 2,..., n}, Aj is the point set of player i € N, the set of decision-making nodes is written as A = (A1; A2,..., An} and Af is the set of terminal points. The path (orbit) of graph < A, 7 > is the sequence of points a0, a1,..., at. For k = 1,..., t,... we have ak € 7 < ak-1 >. An orbit is called a situation, if it is infinite or it contains terminal point a; € Af. All of the rest orbits are called opening situation. Every mapping Sj : Aj ^ A satisfying the condition sj C 7 is called the simple strategy of player i. The sets of all the simple strategies of player i is written as Sj. Situation a € A and simple strategy (s1,..., sn) GS1 x — -x Sn define the situation < a, s1;..., sn >= a, s(a), s2(a),..., where s = S1U '''U sn. If each path on graph < A, 7 > is finite, then every point a € A has a relation with a mapping Fa : S1 x ■ ■ ■ x Sn ^ Af, which maps the situation^,..., sn) under a simple strategy (s1,..., sn) to a terminal point of the situation < a, S1,..., sn >.
Definition 1. Giving every point a an n-dimensional real vector fa = (/a,..., fai)T, it is called a point payoff vector of point a and the i-th component fa is called player
i ’s point payoff of point a.
Definition 2. The n-dimensional vector h(ar , ...,a;) = J^ k=r fak = (h1(ar , ...,a;), ...,hn(ar,...,a;))T is called situation payoff vector corresponding to situation ar, ...,a;(a; € Af) on graph < A,7 >. The i -th component hj(ar ,...,a;),i = 1,..., n is called player i’s situation payoff corresponding to situation ar,..., a;.
According to the definition of simple strategy, different situations may lead to different situations all from initial point ar. Suppose that the situation corresponding with situation (s1,..., sj,..., sn) is ar,..., a;(a; € Af), and the situation corresponding with (s1,..., sj,..., sn) is ar ,...,ak(ak € Af). Notation <j is defined as following:
hj (ar ,...,a;) < hj (ar,..., afc) ^ Far (s1,...,sj ,...,Sn) <j Far (S1,... , sj,..., Sn) where i = 1,..., n, j = 1,..., n.
Choosing a0 € A as an initial point, non-cooperative simple games rao (T) is achieved on graph with point payoff vector T =< A, 7; A1,...,An,Af; fa^A > where the strategy set of player i is Sj, set of terminal point is Af, and fa is point payoff vector of point a on graph < A, 7 >.
3. Existence Theorem of Absolute Equilibrium about Games on Connected Graph with Point Payoff Vector
Theorem 1. Nash equilibrium situation exists under simple strategy for each simple game Tao (T), where ao € A.
Proof. First, the subset of point set is defined for a by induction method as follow:
a) Co = Af,
b) Suppose that for all the ft < a, subset C^ C A have been defined.
If a is finite, then Ca = Ca-1 (a € A\Af : y < a >C Ca-1} is defined. If a is
infinite, then Ca = U^<a C^. The smallest ordinal number p(a) is called the rank of point a € A, we have a € Cp(a). If there isn’t infinite path on graph < A,y >, then every point has rank and the rank is finite.
Second, mapping s* : A ^ A is defined according to the rank of points by induction method as following. For each 0 rank point a(a € Af), we define s*(a) = a. If the rank of point a € A\Af is p(a) = a, and mapping S*(a') is defined by point a' € A with rank p(a') < a. If y < a' >= 0, then s*(a') € y < a' >. Therefore, s* has been defined in Ca-1, and it satisfies condition s* C y. s*a-1 is noted as s*’s restriction which is established in the subset Ca-1 P| Aj (j = 1,..., n). Since Ca-1 is Y-steady, which is to say that the situation whose initial point has emerged into Ca-1 is completely located in Ca-1, and s*a-1 is regarded as a simple strategy of player j in the subgame of Ca-1. Considering a € Ca\Ca-1, we have 0 = y < a >C Ca-1. As pointd above, function Fx (s*a-1,..., sna-1) is defined on each x € y < a >. We denote Ta = (Fx(s*a-1,..., sna-1) : x € y < a >}, namely, Ta is the terminal point set of situation whose initial point passes set y < a > by player j = 1,..., n adopting strategy s*a-1. Obviously, Ta is the nonempty subset of the set of terminal points which can be reached from point a. If a € Aj, we can calculate player i’s situation payoff, the maximum of which is h*, in every situation of subgame Ta. When player i chooses maximal situation payoff h* on point a, and the chosen point is x*. We denote s*(a) = x*. Hence, the following relation is satisfied for each x € y < a >:
Fx(s*a-1,..., sna-1) <j Fx* (s*a-1,..., sna-1) (1)
By induction method, the mapping s* has been defined for each a € A. If Y < a >= 0, then s*(a) € y < a >. Namely, s* C y. If s* is the restriction of mapping s* in Aj, then s* is a simple strategy of player j.
Finally, it is proved by induction method for the rank of a0 that situation s* = (s*,...,sn) is Nash equilibrium of every game rao(T), where a0 € A. Situation s* = (s*,..., sn) is called absolute equilibrium of games on graph < A, y >.
Step 1. Suppose p(a0) = 0 , that is to say a0 € C0 = Af . Now the value a0 of function Fao is independent of the situation. Therefore, every situation is Nash equilibrium.
Step 2. If a is given, we suppose p(a0) = a and situation (s*,..., sn) is Nash equilibrium of every game rx(T), where p(x) < a. The initial point a0 € A is chosen, we will prove that (s 1,..., sn) is Nash equilibrium of game rao(T). In fact, if a0 € Aj, suppose that player j(j = 1,..., n) adopts the simple strategy sj instead of s*. If j = i, considering y < a0 >C Ca- 1, we get the result. We only need to consider j = i. So note s*(a0) = a1, sj(a0) = a'j_. By formula (1), we have
Fa1(s1,
,sn) <j Fa1 (s1,
,sn)
Because a1 € y < a0 >C Ca-1 , then according to assumption mentioned above,
j
Fa1 (s1, • • • , sj-1, sj, sj+1, • • • , sn) < Fa1 (s1, • • • , sn)
By the relations between the two formulas above, we have
Fa1 (s * ,•••, s*-1,sj ^s^ < Fai (s * ,
(2)
s
According to the definition of Fa and s , the following equations are satisfied,
Fao (s1 Fai (s * ,•••,*
TP { „* J „* \ TP { „* J „* \
Fao (s1, • • • , sj-1, sj, sj+1, • • • , sn) = Fa1 (s1, • • • , sj-1, sj, sj+1, • • • , sn)
Considering formula (2), we have
j
Fao (s1, • • • , sj-1, sj, sj+1, • • • , sn) < Fao (s1, • • • , sn)
Namely, situation (s1, • • •, sn) is Nash equilibrium in game Fao (T). The proof of the theorem is finished.
4. Algorithm about Absolute Equilibrium in Games with Point Payoff Vector on Connected Graph
First, calculate the rank p(a) of the point a € A on graph < A, y > according to the definition of rank. Assume maxaG^p(a) = T. The set of points A on graph < A, y > is split up into T + 1 subsets P0, P1, • • •, PT , where Pk is the set of points whose rank equal to k, Ufc=0 Pfc = A, Pl H = 0, l = m. In the following, we will use backward induction method according to the ranks of the points.
Step 0: Consider each point a0 whose rank equals to 0, namely, a0 € P0 = C0 = Af. Since nobody makes move here, by definition 2, we have h(a0) = fao. Denote function by r0 : P0 ^ R, where r0(a0) = hj(a0), let r0(a0) = (r0(a0),•••,rn(a0))T = h(a0), we denote s*(a0) = a0.
Step 1: Consider each point a1 € P1 . Since y' < a1 >C P0 for a1, by definition 2, we have h(a1,a0) = fa1 + r0(a0). Assume a1 € Aj, then player i chooses a0 € y' < a1 > satisfying maxaoG7/<a1>hj(a1, a0) = hj(a1, a0). Denote function by r1 : P1 ^ R, where r1 (a1) = hj(a1, a0) on a1, and let r1 (a1) = (r1 (a1), • • •, rn(a1 ))T = h(a1, a0). Now we get s* (a1) = a0 on a1 € P1.
Step 2: Consider each point a2 € P2. Denote y' < a2 >= Z0 < a2 > (J Z1 < a2 >, where Z0 < a2 >C P0 prescribes the set of points next to a2 with 1 rank. In the following part, we use the similar prescription.
1) For the points in Z0 < a2 >C P0, when a0 € y' < a2 >, by definition 2 we have
h(a2 ,a0) = fa2 + r0 (a0).
2) For the points in Z1 < a2 >C P1, a0 has been chosen on step 1, by definition 2 we have
h(a2,a1,a0) = fa2 + r1 (a1)
When a0 € y' < a2 >, by definition 2 we have h(a2,a0) = fa2 + r0(a0). Assume that a2 € Aj, then player i choose the point a1 € y' < a2 > which can reach
max|maxao6Zo(a2)hj(a2,a0), max01£Z1(a2)hj(a2, a1, a0)} . Also we denote function by r2 : P2 ^ R, where
2( ) = / hj(a2, a1), if a1 € ^0^2) C p0
rj (a2) \ hj(a2,a1,a0), if a1 € ^1^2) C p
at a2, let r2(a2) = (r2(a2), • • •, rn(a2))T. Now we get s*(a2) = a1 at a2 € P2 .
Step t: Consider the each point at € Pt,t < T. Now r0(a0),•••,r*-1 (at-1) and r0(a0), • • • ,rt-1(at-1), have been definite.
Denote
y' < at >= Z0 < at > (J Z1 < a* > (J ■ ■ ■ ^ Z*-1 < a* >
For the points in Z0 < at >C P0, when a0 € y' < at >, by definition 2 we have
h(at,a0) = /at + r0 (a0)
1)For the points in Z1 < at >C P1, when a1 € y' < at >, a0 has been chosen on step 1. By definition 2, we have
h(at,a1,a0) = /at + r1(a1)
t) For the points in Zt-1 < at >C Pt-1 , when at-1 € y' < at > and a0, a1, • • •, at-2 has been chosen on step t — 1, by definition 2 we have
h(at,at-1, at-2, - • • ,60) = /at + rt-1 (at-1)
Assume that at € Aj, then player i chooses at-1 € y' < at > which can reach
max / maxaoeZo(atmaxa1GZ1(at)h(at,a1,60),l • • • , maxat_1 £Zt~1 (at) h(at, at-1, 6t-2, • • • , a0) } J
Denote function by rt : Pt ^ R, where
if ®t-1 € Z0 < at >C P0
if at-1 € Z1 < at >C P1
a0), if at-1 € Zt-1 < at >C Pt-1
at at, let rt(at) = (r1 (at),•••,rn(at))T . Now we get s*(at) = at-1 on at € Pt. Continue the process till t = T. Similarly, we get s* (aT) = aT-1.
Given all above, by the algorithm we get the player’s choice for each point on connected graph with point payoff vector. According to the proof of theorem 3, when we choose arbitrarily point a0 € A to be the initial point on connected graph < A, y > with point payoff vector, strategy (s *,•••, sn) which is independent on a0 is Nash equilibrium on simple game rao (T), that is, situation s * = (s *,•••, sn) is the absolute equilibrium of the game. The equilibrium route is related to the initial point. When we choose the point a0 € PL, 0 < L < T and a0 € Aj as the initial point, the absolute equilibrium s can define the equilibrium route of the simple game rao (T), and the payoff on equilibrium situation is:
hj (at, at-1),
t/ \ 1 hj(at, at-1^0^
rj(at) = ] ••,•••
hj (at, at-1, at-2,
rL (a0) = (rL (a0),•••,rL(a0 ))T
We need to point out that, in the algorithm above, the definition of function rk (a) on some point a E Pk, 0 < k < T maybe more than one, so choose one of them randomly as defined.
5. The Calculation Model of Absolute Equilibrium about Games on Three-dimensional Connected Graph with Point Payoff Vector
Figure 1
Consider the three-dimensional connected graph < A, 7 > (Fig 1), where the set of players is N = {1, 2, 3}. The terminal payoff is
Af = {61, 62, 63, 64, 65, be, 67, bs, 69, bio, bii, bi2, bi3, bi4}
The decision point sets of player 1, player 2, player 3 are respectively Ai = {aooo, aiio, ao20, ai2i, aoii, aoo2}, A2 = {aioo, aoi2, ao2i, am, ai22, aio2}
A3 = {aoio, ai2o, aooi, aioi, aii2, ao22}
Players’ strategy: On graph < A, 7 > , we define the players’ strategy in horizontal and two-dimensional points are along the mesh to the right, the left or terminal points; the strategy of players in right-and-left and two-dimensional points are along the mesh to downward ,the front, or terminal points; the strategy of players in fore-and-aft and two-dimensional points are along the mesh to the right, downward or terminal points.
By definition 1, we give the point payoff vector on every point on graph < A, 7 >. Then we get the game on three-dimensional mesh-like and connected graph with point payoff vector T =<A,y; Ai ,A2,A3, Af; faeA >. This example gives the point payoff vector as following:
faooo = (1, 2, 2)T,fa010 = (2, 1, 3)T, fao20 = (4, 2, 1)T, faooi = (5, 3, 2)T
faoii = (2, 2, 2)T,fao21 = (1, 3, 2)T,faoo2 = (2, 4, 5)T,fao12 = (6, 5, 2)T
fao22 = (1, 5, 3)T,faioo = (3, 6, 7)T,fano = (2, 5, 1)T,fai2o = (5, 7, 4)T
faioi = (2, 5, 4)T, faiii = (4, 0, 2)T,fai2i = (3, 3, 4)T,faio2 = (4, 6, 3)T
faii2 = (0, 5,3)T,fai22 = (3,2,4)T,fbi = (4, 3,2)Tf = (2,5,4)T
fb3 = (3,4,5)Tf = (4,6, 3)Tf = (2, 5,3)Tf = (4,2, 3)Tf = (3, 2,6)T
fb8 = (4,3,1)T, fb9 = (3, 5,7)T, f6io = (4, 3,5)T, f6n = (2,6, 5)T, fb^ = (3,2,4)T
fbi3 = (2,4,1)T, fbi4 = (5,3, 2)T
First, compute the ranks of all points. Get the rank-subdivision of the point set A on the graph < A, 7 >:
Po = {6i ,62,63,64,65,66,67,6s ,69, 6io, 6ii, 6i2 ,6i3,6i4 },Pi = {ai2o}
P2 = {aiio,ai2i,ao2o},P3 = {aioo,aiii, aoio, ai22,ao2i}
P4 = {aioi,aii2,aooo, aoii, ao22}, P5 = {aio2,aooi ,aoi2},P6 = {aoo2}
Step 0: Consider every point in 0 rank point set Po. According to algorithm, we note ro (6j) = h(6j), j = 1,..., 14, and define s*(bj) = bj, j = 1,..., 14.
Step 1: Consider every point in 1 rank point set Pi = {ai2o}. According to algorithm, we get h(ai2o, 6i2) = fai2o + ro (6i2) = (8,9, 8)T h(ai2o ,6i3) = (7,11, 5)T. Since ai2o E A3, h3(ai2o, bi2) = 8 > 5 = h3(ai2o,6i3), player 3 will choose bi2, noting ri(ai2o) = (ri(ai2o),ri(ai2o(ai2o))T = h(ai2o, 6i2) = (8, 9,8)T, similarly we get s*(ai2o) = 6i2.
Step 2: Consider every point in 2 rank point set P2 = {aiio ,ai2i ,ao2o}. For aiio E P2, we have 7' < aiio >= Zo < aiio > U Zi < aiio >, where Zo < aiio >= {6io,6ii},Zi < aiio >= {ai2o}. We get h(aiio,bio) = (6,8,6)T, h(aiio,bii) = (4,11, 6)T,h(aiio,ai2o,bi2) = faiio + ri(ai2o) = (10,14, 9)T. For aiio E Ai, and max{hi(aiio, bio), hi(aiio,bii),hi(aiio, bi2o, bi2)} = 10 = hi(aiio, ai2o,bi2), So player1 will choose ai2o , noting r2(aiio) = (r2(aiio),r|(aiio),r2(aiio))T = h(aiio, ai2o, bi2) = (10,14,9)T.
Then we get s*(aiio) = ai2o.
Similarly, for ai2i E P2, since ai2i E Ai, and player 1 have the only choice ai2o, we note r2(ai2i) = (r2(ai2i),r|(ai2i),r2(am))T = h(am ,ai2o A2) = (11,12,12)T, and get s*(am) = ai2o. For ao2o E P2, we have 7' < ao2o >= Zo < ao2o > (J Zi < ao2o >, where Zo < ao2o >= {bi4},Zi < ao2o >= {ai2o}. Since ao2o E Ai ,hi(ao2o ,bi4) =9 < 12 = hi(ao2o ,ai2o ,bi2), player 1 will also choose ai2o, noting r2(ao2o) = (r2(ao2o),r2(ao2o),r2(ao2o))T = h(ao2o,ai2o,bi2) = (12,11,9)T, then we get s*(ao2o) = ai2o.
Step 3: Consider every point in 3 rank point set P3 = {aioo,aiii, aoio,ai22,ao2i}. The results are given as following: r3(aioo) = (13,20,16)T, s*(aioo) = aiio, r3(am) = (14,14,11)T, s*(am) = aiio,r3(ai22) = (14,14,16)T, s*(ai22) = ai2i; r3(ao2i) = (12,15,14)T, s*(ao2i) = am. For aoio E P3, we have 7' < aoio >= Zo < aoio > (jZi < aoio > UZ2 < aoio > where Zo < aoio >= Zi < aoio >= 0, Z2 < aoio >= {ao2o,aiio}. Since aoio E A3, and h3(aoio,ao2o,ai2o,bi2) = 12 = h3(aoio, aiio, ai2o,bi2), by assumption, player 3 can arbitrarily choose aiio or ao2o. Then if player 3 choose point ao2o, noting r3(aoio) = (14,12,12)T, then s*(aoio) = ao2o.
Step 4: Consider every point in 4 rank point set P4 = {aioi, aii2, aooo, aoii, ao22}. The results are, r4(aioi) = (15, 25,20)Ts*(aioi) = aioo,r4(aii2) = (14,19,19)T, s*(aii2) = ai22,r4(aooo) = (15,14,14)T ,s*(aooo) = aoio;r4 (aoii) = (16,14,14)T, s*(aoii) = aoio,r4(ao22) = (15,19,19)T ,s* (ao22) = ai22.
Step 5: Consider every point in 5 rank point set P5 = {aio2, aooi, aoi2}. The results are, r5(aio2) = (19,31, 23)T, s*(aio2) = aioi,r5(aooi) = (20, 28,22)T, s*(aooi) = aioi; For aoi2 E P5, since aoi2 E A2, and h2(aoi2, ao22, ai22, ai2i, ai2o,bi2) = 24 = ^2(aoi2, aii2, ai22, ai2i, ai2o, bi2), player 3 can arbitrarily choose aii2 or ao22. If player 3 chooses point ao22, noting r5(aoi2) = (21, 24,21)T, then s*(aoi2) = ao22. Step 6: Now consider the unique 6 rank point aoo2 E P6. We have 7' < aoo2 >= Za o < aoo2 > Zi < aoo2 > Z2 < aoo2 > Z3 < aoo2 > Z4 < aoo2 > IJZ5 < aoo2 >, where Zo < aoo2 >= {bi,b2,b3}Zi < aoo2 >= Z2 < aoo2 >= Z3 < aoo2 >= Z4 < aoo2 >= 0, Z5 < aoo2 >= {aio2,aooi,aoi2}. For aoo2 E Ai, noting r6(aoo2) = (23,28, 26)T, we get s*(aoo2) = aoi2.
Finally we get the absolute equilibrium of the game, which is noted by bold black line in Fig.1.
6. Conclusion
The result of this paper is valid for random finite connected graph, no matter it is two-dimension or three-dimension. Of course the result includes the game tree we have known. We choose three-dimensional mesh-like graph in this paper only because initial inspiration comes from the exception for carrying out game’s research on three-dimension space. Moreover, on usual dynamic games, usually the absolute equilibrium is perfect equilibrium. But from a different aspect, actually the absolute equilibrium is stronger than the perfect equilibrium.
The inductive method of the rank of point on the graph used in this paper will show its power on the research of game on complex graph. For simple structure graph, by listing all likely appeared situations, we can always transform game graph to the game tree. Then we can solve it by common method. However, considering games on three-dimensional connected graph, the complex computation leaded by the large amounts of situations can hinder and conceal some important research for the nature of game. We have reasons to believe that the arithmetic of the absolute equilibrium about the game for limited connected graph established by this paper can be popularized to partial cooperation game of finite three-dimensional graph with variable coalitional structure and so on.
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