CC BY
3Contributions to Game Theory and Management, VIII, 252—267
Multistage Game Model with Time-claiming Alternatives *
Ovanes L. Petrosian1 and Levon K. Babadzhanjanz2
1 St.Petersburg State University,
Faculty of Applied Mathematics and Control Processes, Universitetskii pr. 35, Petergof, St.Petersburg, 198504, Russia E-mail: [email protected]
2 St.Petersburg State University,
Faculty of Applied Mathematics and Control Processes, Universitetskii pr. 35, Petergof, St.Petersburg, 198504, Russia E-mail: [email protected]
Abstract The new model of multistage game with perfect information, on a closed time interval is considered. On each stage of the game player chooses one of the alternatives and time to perform them. The payoffs depend upon trajectory and the time, at which game terminates. As a solution of this game subgame perfect e - Nash equilibrium is taken.
Keywords: Perfect information, Nash equilibrium, Time-claiming alternative.
1. Introduction
The following finite stage game with perfect information is considered. In each vertex of the game tree belonging to the set of personal moves of player the finite number of basic alternatives is fixed and for each given basic alternative a closed time interval is defined. The elements of this time interval are interpreted as time instants at which the basic alternative can be realized in a given vertex. Each basic alternative in the multistage game with Time-claiming alternatives is associated with an infinite number of alternatives, the basic alternative with corresponding time values we shall call bunch of alternatives.
As usual the strategy of player is a mapping which corresponds to each vertex from the set of personal moves of the player the pair consisting from the index of basic alternative and time necessary to realize this alternative. If the n-tuple of strategies is chosen by players the trajectory of the game path can be uniquely defined. This path consists from the sequence of basic alternatives and corresponding time parameters chosen by players. Payoff function of player for each trajectory of the game continuously depends upon the time when the game terminates and it is a uniformly bounded function. However it is proved that payoff function of the player not necessary continuously depends upon his strategy (part of his strategy, time at which the alternative must be perform). This makes impossible the existence of subgame perfect Nash equilibrium. The example of this case is presented and the existence of subgame perfect e - Nash equilibrium is proved.
This type of games arises in game-theoretical modeling of many real life and business situations.
* The authors acknowledge Saint-Petersburg State University for a research grants No.9-37-345-2015 and 9-38-205-2014.
2. Difference between classical multistage game with perfect
information and multistage game with Time-claiming alternatives
Description of multistage game with Time-claiming alternatives has some differences from multistage game with perfect information.
Multistage game with perfect information is defined on the tree like graph. Denote the game as r and graph as G = (X, F), where X is a finite set of vertices and F is a multivalued mapping from X to X (Vz G X, Fz C X ). Consider a partition of the set of vertices X :
Xi • • • Xn, X„+1, X = U"+11Xi, Xfc n X, = 0, k = l,
where Fz = 0 for z G Xn+1. Set Xi; i = 1 • • • n is a set of personal moves of player i and set Xn+1 is a set of final positions. On the set of final positions Xn+1 payoffs H1(z) • • • Hn(z), z G Xn+1 are defined. We call a strategy of player i, mapping u which to each position z G Xj uniquely correspond the position y G Fz. Denote the set of all possible strategies of player i as Uj. We call an ordered set u = («1 ••• Uj ••• un), where u G U the situation in the game r. Define the payoff function Kj, for each player i = 1 • • • n in the game r as follows:
Kj(«1 • • • Uj • • • u„) = Hj(z;), i = 1 • • • n,
where z; G Xn+1 is a final position which corresponds to the situation («1 • • • uj • • • un) in the game r. Function Kj, i =1 • • • n is defined on the set of situations U = nn=1 Uj. Thus multistage game with perfect information has the following form
r = (N, |Uj}j£N, |Kj}j£N),
where N is a set of players.
Multistage game with Time-claiming alternatives is defined on the tree like extended form of the graph G. Denote the extended tree like graph as G(Y, where Y is the set of positions and Y = X x [t0,T], where X is the set of positions from the graph G and [t0,T] is a closed time interval, in which the game takes place. <P is a multivalued mapping from Y to Y (Vz G Y, C Y), = Fz x [t, T], where z = (z,t), z G X, t G [t0,T].
Denote a pair of positions (z, z'), where z = (z,t), z' = (z',t'), z' G = Fz x [t; T] for any possible pair of values t,t' (Vt' G [t,T], Vt G [t0,T]), as an arc p = (z, z'). Denote a sequence of arcs p = (p1,p2 • • • Pfc • • • ), where p1 = (z 1, z1), P2 = (z2,z2) • • • pfc = (zfc,z'k), • • • and z1 = z2, z2 = z3 • • • z'k = zfc+1 • • • for any possible values tj, tj (Vtj G [tj, T], Vtj G [to, T]), i, j = 1, 2 • • • k • • •, as a path in the graph r. In the multistage game with Time-claiming alternatives the length l(p) of the path p = (p1,p2 • • • pk) is a number of arcs in the path or number of different positions z minus one from graph r, l(p) = k. Consider a partition of the set of positions Y:
Y1 • • • Yn, Y„+1, Y = un+11 Yj, Yfc n Y = 0, k = l,
where = 0 for z G Yn+1. Set Y = Xj x [t0,T] is the set of personal moves of player i and Y^+1 = Xn+1 x [t0,T]. On the set of final positions Y^+1 payoffs H 1(z) ••• Hn(z), z = (z,t) G Yn+1 are defined. Functions Hj(z) = Hj(z,t), i = 1 • • • n continuously depend upon the parameter t and are uniformly bounded functions on the closed interval [t0,T]. We call strategy of player i mapping uj
which to each position z G Y correspond the position z' G . Denote the set of all possible strategies of player i as UWe call an ordered set u = (u1 • • • U • • • Un), where Uj G Uj, situation in the game r. Define the payoff function Kj for each player i = 1 • • • n in the game r, as follows:
Kj(Ui • • • Uj • • • U„) = Hj(z;),i = 1 • • • n,
where z; G Yn+1 is a final position which corresponds to the situation (U1 • • • Uj • • • Un) in the game r. Function Kj, i = 1 • • • n is defined on the set of situation U = nn=1 Uj. The length of all paths in the game r is uniformly bounded, because the set of positions is finite. We call the length of the game r the length of the longest path in the game r. By construction the lengths of the games r and r are the same.
3. Definition of multistage game with Time-claiming alternatives
The multistage game with Time-claiming alternatives has the following normal form:
r =(N, {Uj|j£N, {Kj|j£N)
Suppose all paths in the game r have the same length l, then the game proceeds as follows:
1. Let z0 = (zo,to) G Yj1 = Xj1 x [io,T] then player i1 chooses
Z1 = (z1,t1) G = FZ0 x [to, T], t1 < T
2. if z1 = (z1,t1) G Yj2 = Xj2 x [t0,T] then player i2 chooses
Z2 = (Z2,t2) G = Fzi X [t1,T], t2 < T
k. if zk-1 = (zfc_1,ifc_1) G Yjk = Xjk x [to, T] then player ifc chooses zfc = (zk,tfc) G ^Zk-i = Fzfc_i x [tfc_1,T], tk < T
l. if z;_1 = (zl_1,t;_1) G Yj; = Xjl x [to, T] then player i; chooses z; = (z;, t;) G = FZ;-1 x [t;_1,T] (FZ;-1 C Xn+1) and the game terminates.
The game can terminate if player i chooses z = (z, t), where z G Xn+1 (where t is any instant from the closed interval [to,T])
4. Existence of subgame perfect e - Nash equilibrium in multistage game with Time-claiming alternatives
The following theorem can be proved.
Theorem 1. In any multistage game r there exists a subgame perfect e-equilibrium u*
It is shown in the example that in multistage game with Time-claiming alternatives sometimes is impossible to construct a subgame perfect equilibrium, because of noncontinuous dependence of payoff function of players upon time (part of the
strategy). During the backward induction process in multistage game with perfect information on each stage player is maximizing his payoff choosing the basic alternative and in this game model player in addition chooses time to perform this basic alternative. Consider subgame r, t) (z G X,t G [t0,T]). Suppose that in this subgame there exist a subgame perfect Nash equilibrium u^- t) = (u^ t) 1 • • • u^ t) n). Let Kj(z, t; u^ ^ 1 • • • u^ ^ n), i G N be the payoff function of player i in this Nash equilibrium. Since the NE u^- ^ is fixed this function depends only on initial conditions of the subgame, and we can write
Vi(z,t) = Ki(z,t;it),1 • •• ,t),J,« e
Functions Vi (z,t),i G N we shall call value functions. Also introduce the following function
V(zfc,tfc)(t) = max {Vi(z,t),t e [tfc,T)} = V(z,t), where xfc e Xi
Vj (zk ,tfc)(t) = Vj (z,t),j = «
Vi(zfc,tfc)(t) = Hi(xfc,tfc)vt e [tfc,T], if xfc e x„+i,i e N
Vi(zk,T)(t) = Hi(zfc,t),i e N
Vi (zk,tk )= sup Vi(zk ,tk )(t) (1)
tefifc, t ]
Problem takes place when player chooses time to perform basic alternative, because the value in the game r (Vi(zk, tk)(t)) sometimes cannot be reached (see example). Function Hj(z, t),z G Xn+1 continuously depends upon the parameter t and is uniformly bounded in the closed interval [t0,T] ^ [tj,T]. Then function Vi(zk,tk)(t) is also uniformly bounded on the closed interval and sup Vi(zk,tk )(t)
_ tefifc, t ]
exists. Therefore in the game r we can use subgame perfect e - Nash equilibrium as a solution in the game.
5. Example
Consider an example from (Papayoanou, 2010). Game r takes place on the graph G = (X, F) below (see Fig. 1). The set of players N consists of two players, N = {Alpha, Beta}. On the first stage of the game r, in the position z0 player Alpha has two alternatives z1,z3 (Accept or Reject financial proposal from player Beta). If player Alpha chooses the alternative z3 then the game terminates, if player Alpha chooses the alternative z1 then the game proceeds and player Beta makes his move. On the second stage of the game r, in the position z1 player Beta has two alternatives z2, z6 (Offer to Alpha a better financial proposal or Compete with player Alpha). If player Beta chooses the alternative z6 then the game terminates, if player Beta chooses the alternative z2 then the game proceeds and player Alpha again makes a move. On the third stage of the game r, in the position z2 player Alpha has two alternatives z4, z5 (Accept the better financial proposal from player Beta or Compete with player Beta). In either cases the game terminates, payoffs are defined for both players at the end of the game (see Fig. 1).
Fig. 1: Example
Often it is important to determine the time necessary to make a decision. Consider related example, which was proposed using the approach of multistage games with time claiming alternatives. In this example it is possible to simulate more properly and efficiently this economic situation, because the model of Multistage game with Time-claiming alternatives considers also a time necessary to make a decision (move). Game r takes place on the graph G = (Y,below (see Fig. 2). The set of players N has not changed. On the first stage of the game r, in the position zo = (zo, to) player Alpha has two bunches of alternatives z1 = (z1, t1),z3 = (¿3, ts), where t1, t3 G [to, T]. In this model player Alpha is not just choosing the alternative Accept or Reject financial proposal from player Beta, but also he is selecting the time instant to choose the alternative (basic alternative). If player Alpha chooses the alternative z3 = (23, ts) then the game terminates at the moment of time t3 and player Alpha gets the payoff
H
Alpha
(*3,ts)
160 1 + t3 '
player Beta gets the payoff
HBeta(z3513)
280 1+ t3
If player Alpha chooses the alternative zi = (zi,ti) then the game proceeds and player Beta makes a move. On the second stage of the game r, in the position zi = (zi, ti) player Beta has two bunches of alternatives Z2 = (z2, t2), zg = (zg, tg), where t2,t6 € [ti,T] (Offer to Alpha a better financial proposal or Compete with player Alpha and choose time instant to make it). If player Beta chooses the alternative z6 = (zg, tg) then the game terminates at the moment tg and player Alpha gets the payoff
HAlpha^tg) = ln(tg + 1) - 25,
player Beta gets the payoff
HWze,te) = -20(t6 - 4)2 + 730.
If player Beta chooses the alternative z2 = (z2,t2) then the game proceeds and player Alpha again makes his move. On the third stage of the game r, in the position z2 = (z2, t2) player Alpha has two bunches of alternatives z4 = (z4, t4), z5 = (z5,t5), where t4,t5 G [t2,T] (Accept the better financial proposal from player Beta or Compete with player Beta and choose time instant to make it). In either cases the game terminates. If player Alpha chooses the alternative z4 then he gets the payoff
HAIpha (z4,t4) = -0, 35(t4 - 6)2 + 204,
player Beta gets the payoff
HBeta (z4,t4) = -¿42 + 690. If player Alpha chooses the alternative z5 then he gets the payoff
HAIpha(z5,t5) = -(t5 - 6)2 + 209, player Beta gets the payoff (see Fig. 2)
HBeta (z5,t5) = -t52 +410.
Fig. 2: Example
Payoff functions depending upon the time on which the game r terminates in this example have the following form:
We use standard backward induction procedure to find the solution of the game. The backward induction in this game is based on the length of the game (which is a number of arcs in the maximal path l = 3). We have to solve subgames r(Ziiti), zi S X for any starting time ti S [0,10]. Denote by VAipha(k,zi,ti) and VBeta(k,zi,ti), values of subgames r^Zi,ti) with length k for player Alpha and Beta (the exact upper bound of values of players payoffs when a fixed subgame perfect e - Nash equilibrium is used).
Consider subgame r(Z2,t2), where in the position = (z2, t2) player Alpha makes a move. z2 is a starting position of this subgame and t2 S [0,10] is a time chosen by the player Beta on the previous stage of the game r (in the position ~z1 = (zi, ti)). In this subgame player Alpha chooses between two bunches of alternatives z4 = (z4, t4), z5 = (z5,t5) (&Z2 = {z4,z5}). Since the positions z4,Z5 S Yn+1 and t4,t5 are time instants, when the game r terminates the payoffs are defined in this positions and depend upon t4,t5 and are equal to HAipha(z4,t4), HAipha(z5,t5) (see Fig. 4).
Fig. 4: Example
Since it is impossible for player Alpha to choose a moments of time t4, < t2, because the subgame rz2;t2 starts at the moment of time t2 we must construct a solution for the player Aplha for each possible starting time t2. The value of the subgame r(z2,t2) for player Alpha is
VAipha(1,z2,t2) =max{ sup (-0, 35(t4 - 6)2 + 204); sup (-(¿5 - 6)2 + 209)}.
Î4£[Î2,i0] Î5£[Î2,i0]
Consider different cases, where t2 € [0; 6], t2 € (6; 8, 8], t2 € (8, 8; 10]:
1. t2 € [0;6], where t = 6 is determined from the condition HAipha(z4;6) =
sup HAipha (z4, t) then the optimal behavior of player Alpha in position
te[0,i0]
z2 = (z2,t2) is defined by formula UA;pha(z2,Î2) = (z5; Î5), where Î5 = 6. VAipha(1,z2,t2) = max{ sup (-0, 35(t4 - 6)2 + 204); sup (-(Î5 - 6)2 +
Î4£[Î2,i0] Î5é[Î2,10]
+209)} = sup (-(t5 - 6)2 + 209) = -(6 - 6)2 +209 = 209
Î5£[Î2,10]
VBeta (1,z2,t2) = -62 +410 = 374 (Since in the position z2 = (z2,t2) player Beta is not making a move)
2. t2 € (6; 8, 8], where t = 8, 8 is determined from the condition HAipha(z4; 8, 8) = HAipha(z5; 8, 8) then the optimal behavior of player Alpha in position =
(z2,t2) is defined by formula (z2; t2) = (z5; t5), where t5 = t2.
VAipha(1,z2,t2) = max{ sup (-0, 35(t4 - 6)2 + 204); sup (-(t5 - 6)2 +
Î4É[Î2,10] Î5£[Î2,10]
+209)} = sup (-(t5 - 6)2 + 209) = -(t2 - 6)2 + 209)
Î5£[Î2,10]
VBeta(1,z2,t2) = -t2 +410
3. t2 € (8, 8; 10] then the optimal behavior of player Alpha in position z2 = (z2, t2) is defined by formula WAipha(z2;t2) = (z4; t4), where t4 = t2.
VAipha(1,z2,t2) =max{ sup (-0, 35(t4 - 6)2 + 204); sup (-(t5 - 6)2 +
Î4É[Î2,10] Î5£[Î2,10]
+209)} = sup (-0, 35(t4 - 6)2 + 204) = -0, 35(t2 - 6)2 +204
Î4É[Î2,10]
VBeta (1,z2,t2) = -t2 +690
Introduce the following function:
VAipha(1, z2, t2)(t) = max{-0, 35(t4 - 6)2 + 204; -(t5 - 6)2 + 209,
where t € [t2, 10]},
it is easy to see that sup VA;pha(1, z2, t2)(t) = te[t2,i0]
= sup [max{-0, 35(t4 - 6)2 + 204; -(t5 - 6)2 + 209, where t € [t2,10]}] =
te[t2,i0]
= max{ sup (-0, 35(t4 - 6)2 + 204); sup (-(t5 - 6)2 + 209)} =
Î4É[Î2,10] Î5£[Î2,10]
= VAipha (1, z2, t2),
where t corresponds to the moments of time t4 or t5: if VAlpha(1,z2,t2)(t) = -(t - 6)2 + 209 then t corresponds to t5, if VAlpha (1, z2, t2)(t) = -0, 35(t - 6)2 + 204 then t corresponds to t4. On the Fig. 5 we can see function VAlpha(1, z2, t2)(t), where t2 = 0, if subgame r(z2,t2) started at the moment of time t2 = 0. If the subgame r(z2it2) starts for example at the moment of time t2 = 5, then we must consider all values of function VAlpha(1,z2,t2)(t) on the Fig. 5 for t € [5,10]. In this sense the function on the Fig. 5 corresponds to the solution constructed earlier.
Fig. 5: Example
Consider subgame r(-1;tl), where in the position Z1 = (z1,t1) player Beta makes a move. zi is a starting position and ti G [0,10] is a time chosen by the player Alpha on the previous stage of the game r (in the position Z0 = (zo,to)). In this subgame player Beta chooses between the alternatives Z2 = (z2,t2), Z6 = (z6,t6) (&-1 = {Z2,Z6}). Since position Z6 G Yn+1 and t6 is the time instant, when the game r terminates the payoff in this position is equal to Hgeta (z6,t6). Z = (z2,t2) / Yn+i, which means that if player Beta in the position Z1 = (z1,t1) chooses a basic alternative z2 G X and any time instant t2 G [t1,T], then player Beta must take in account which alternative is going to choose player Alpha on the next stage in the position Z = (z2, t2 ) for each moment of time t2 and calculate his payoff accordingly, VBeta(1,Z2,t2) (see Fig. 6).
The concept of Nash equilibrium obtained by backward induction suggests that on the first step of the subgame player who makes a move on this stage expects that in all subsequent subgames players use a fixed predetermined Nash equilibrium. In our case it means that player Beta in the subgame r(-2,t2) expects that player Alpha in the following subgame uses Nash equilibrium strategy, i.e. he expects to get the payoff VAiPha(1,z2,t2) (which is a result of using Nash equilibrium strategy by the player Alpha).
t = 4
......„......„...../s / \
XvV> X
/ X
\
/ \
/ \
Pfleta(l.Z2,t) \
---■ — ----■ « --■-■ -
\
\
\
\ \
0 0 0,5 1,0 1,5 2,0 2,5 3,0 !,!) 4,0 4,5 5,0 5,5 6,0 6,5 7,0 7,5 8,0 8,5 9,0 9,5 10,0
Fig. 6: Example
The decision of player Beta in this subgame depends upon the time ti. The value of the subgame r(z1,tl) for player Beta is
Vseia, (2, zi, ti) = max{ sup VB eta (1,z2,t2); sup (-20(te - 4)2 + 730)}.
i2£[ii,io] t6e[ti,io]
Consider different cases, where ti E [0; 4], ti E (4; 8, 8],ti E (8, 8; 10]:
1. ti E [0;4], where t = 4 is determined from the condition HBeta(z6;4) =
max HBeta(z6,t) then the optimal behavior of player Beta in position zY =
te[o,io]
(zi,ti) is defined by formula UBeta(zi,ti) = (z6; t6), where t6 = 4. Then subgame r(zi,tl) terminates, because z6 E Yn+i.
VBeta(2,zi,ti) = max{ sup VBeta(1, z2, t2); sup (-20(t6 - 4)2 + 730)} =
i2£[ii,i0] i6£[ii,i0]
= sup (—20(t6 - 4)2 + 730) = -20(4 - 4)2 + 730 = 730
t6 e[ti,io]
V4/pha(2,zi,ti) = 1n(4 + 1) - 25 = -23,4
2. ti E (4; 8, 8] then the optimal behavior of player Beta in position zi = (zy, ti) is defined by formula UBeta(zi; ti) = (z6; t6), where t6 = ti. Then subgame r(zi,ti) terminates, because Z6 E Yn+i.
VBeta (2,zi,ti) = max{ sup VBeta(1,Z2,t2); sup (-20(t6 - 4)2 + 730)} =
i26[ii,i0] t6£[ti,i0]
= sup (-20(t6 - 4)2 +730) = -20(ti - 4)2 + 730
t6£[ti,i0]
V4/pha (2, zi, ti) = 1n(ti +1) - 25
3. t1 G (8, 8; 10] then the optimal behavior of player Beta in position z1 = (z1,t1) is defined by formula uBeta(z1; t1) = (z2; t2), where t2 = t1. Then on the next stage UAlpha (z2; t2) = (z4; t4), where t4 = t2.
VBeta (2, z1, t1) =max{ sup VBeta(1,z2,Î2); sup (-20(t6 - 4)2 + 730)} =
t2£[ti,10] tae[ti,10]
= sup VBeta (1, z2,t2) = VBeta (1, z2, ¿1 ) t2£[ti,10]
VAipha(2, z1,t1) = VAipha(1,z2,t0 = -0, 35(t1 - 6)2 + 204
The solution here is constructed using the solution for the player Alpha on the previous step. Consider the following function:
VBeta(2, z1,t1)(t) = max{VBeta(1,z2,t); —20(t - 4)2 + 730, where t G [tu 10]},
it is easy to see that sup VBeta(2, z1, t1)(t) = VBeta(2,z1,t1), te[ti,10]
where t is a moment of time t2 or t6: if VBeta(2, z1, t1)(t) = VBeta (1,z2,t) then t corresponds to t2, if VBeta(2, z1,t1)(t) = -20(t - 4)2 + 730 then t corresponds to t6. On the Fig. 5 we can see function VBeta(2, z1, t1)(t), where t1 = 0, if subgame r(Zl,tl) started at the moment of time t1 = 0. If the subgame r(Zl,tl) starts for example at the moment of time t1 = 5, then we must consider all values of the function VBeta(2, z1,t1)(t) on the Fig. 5 for t G [5,10]. In this sense the function on the Fig. 5 corresponds to the solution constructed earlier.
Fig. 7: Example
It is shown in the solution for the player Beta that the function VBeta(2, z1, t1)(t), where t is a time instant chosen by the player Beta (t2, t6) noncontinuously depends
upon the parameter t. Which means that if subgame rzi,tl starts at the moment of time ti € (6, 4; 10] then
VBeta (2,zi,ti ) = max{ sup Vb eta (1,z2,t2); sup (-20(te - 4)2 + 730)}
t2e[ti,io] te e[ti,io]
can not be reached.
Consider subgame r(z0,t0) = r, where in the position z0 = (z0,t0) player Alpha makes a move. z0 is a starting position and t0 = 0 is a starting moment of the whole game. In this subgame player Alpha chooses between two alternatives zi = (zi,ti), z3 = (z3,t3) (&z0 = {zi,z3}). Since the position z3 € Yn+i, t3 is the time instant, when the game r terminates the payoff in this position is defined and is equal to HApiha(z3,t3). zi = (zi,ti) / Yn+i, this means that if player Alpha in the position z0 = (z0,t0) chooses a basic alternative zi € X and any time instant ti € [t0, T], then player Alpha must determine which alternative is going to choose player Beta z2 or z6 on the next stage and the player Alpha on the stage after z4 or z5. In the position zi = (zi,ti) for each moment of time ti player Alpha can calculate his payoff according to the optimal behavior of player Beta on this stage and optimal behavior of player Alpha on the next one, VAiPha(2,zi,ti) (see Fig. 6) (see Fig. 8).
Fig. 8: Example
if player Alpha chooses the alternative zi = (zi,ti) (uAipha(z0; t0) = (zi; ti)), where ti € (6, 4; 8, 8] then on the next stage player Beta chooses the alternative z2 = (z2,t2) (ÛBeta(zi; ti) = (z2; t2)), where t2 € [ti; 10]. Suppose t2 = 8,8 + e (where e is sufficiently small). Then on the stage after player Alpha chooses the alternative z4 = (z4,t4) (uAipha(z2; t2) = (z4; t4)), where t4 = t2 =8, 8 + e. The
Multistage Game Model with Time-claiming Alternatives value of the subgame r(z0jt0) = r for player Alpha is
VAipha(3, zo,to) = max{ sup ( ); sup VAipha(2, zi, ti)} = t3e[o,io] 1+tie[o,io]
= sup VAipha(2, zi,ti)= VAipha(2,zi, 8, 8) =
ti e[o,io]
= -0, 35((8, 8 + e) - 6)2 + 204 = 201, 2 - e.
Supremum is not reached, although it is possible to choose t close enough to 8, 8 (t + e, where e is sufficiently small) in which value of the function is close enough to the supremum.
The value of the subgame r(z0jt0) for player Beta is
VBeta(3, zo, to) = Vßeta (2, Zi; 8, 8) = Vßeta (1, Z2, 8, 8) = 612, 6 - e
The solution here is constructed using the solution for the player Alpha and Beta on the previous steps. Consider the following function:
VAipha (3, zo,to)(t) = max{ l+t ; V4ipha (2,Zi,t), where t G [0,10]},
it is easy to see that sup VAIpha(3, zo,to)(t) =
te[o,io]
= sup [max{-160-; V^pha(2, zi,t), where t G [0,10]}] =
te[o,io] 1+t
= max{ sup L 160 ); sup VAipha (2,zi,ti)} = V4ipha(3, zo, to),
t3£[o,io] 1+13 ti£[o,io]
where t is a moment of time ti or t3: if VAipha(3, zo,to)(t) = VAipha(2, zi,t) then t corresponds to ti, if VAipha(3, zo,to)(t) = then t corresponds to t3. Function VAipha(3,zo,to)(t) has the following form (see Fig. 9).
Fig. 9: Example
It is shown in the solution for the player Alpha that the function VAipha(3,zo,to)(t), where t is a time instant chosen by the player Alpha (it can be t3 or ti depends on which basic alternative is chosen) noncontinuously depends
upon the parameter t and VAipha(3, z0,t0) = sup VAipha(3, z0,t0)(t) cannot be
te [0,10]
reached.
Subgame perfect e-Nash equilibrium u* has the form:
Aplha = (uAplha[Z0,t0),uAplha(z2 ,t2)) = ((zi;8, 8 + e), (Z4] 8, 8 + e));
UBeta = UBeta(z1 ,t1) = (z2;8, 8 + e)
The payoffs in Subgame perfect e-Nash equilibrium u*
KApiha = 201, 2 - e'; KBeta =612, 6 - e'
where e' is sufficiently small.
6. Conclusion
The model of multistage game with time claiming alternatives can be successfully used in the business or science applications where time is a decisive parameter in the decision making process.
References
Petrosyan, L., Zenkevich, N. (1996). Game Theory. World Scientific Publisher. Papayoanou, P. (2010). Game Theory of Business. Probalistic Publishing. Reinhard, S. (1998). Multistage Game Models and Delay Supergames. Theory and Decision, Springer, 44, 1-36.
Nash, J. (1951). Non-cooperative Games. Annals of Mathematics, 54, 284-295. Kuhn, H. (1953). Extensive games and problems of information. Contributions to the theory of games, 2, 193-216.
