Dynamic Consistency in Discrete-Time Cooperative Games

Abstract

In some economic situations, the economic process is in discrete time rather than in continuous time. The discrete-time counterpart of differential games are known as dynamic games. Bylka et al (Ann. Oper. Res. 97:69–89, 2000) analyzed oligopolistic price competition in a dynamic game model. Wie and Choi (KSCE J. Civ. Eng. 4(4):239–248, 2000) examined discrete-time traffic network. Beard and McDonald (Ann. Int. Soc. Dyn. Games 9:393–410, 2007) investigated water sharing agreements and Amir and Nannerup (J. Bioecon. 8:147–165, 2006) considered resource extraction problems in a discrete-time dynamic framework. Yeung (Ann. Oper. Res. doi:10.1007/s10479-011-0844-0, 2011) examined dynamically consistent collaborative environmental management with technology selection in a discrete-time dynamic game framework. The properties of Nash equilibria in dynamic games are examined in Basar (J. Optim. Theory Appl. 14:425–430, 1974; Int. J. Game Theory 5:65–90, 1976). The solution algorithm for solving dynamic games can be found in Basar (IEEE Trans. Autom. Control AC-22:124–126, 1977; In: Differential Games and Control Theory II, pp. 201–228, 1977). Petrosyan and Zenkevich (Game theory. World Scientific, Singapore, 1996) presented an analysis on cooperative dynamic games in a discrete time framework. The SIAM Classics on Dynamic Noncooperative Game Theory by Basar and Olsder (Dynamic noncooperative game theory, 2nd edn. Academic Press, London, 1995) gave a comprehensive treatment of discrete-time noncooperative dynamic games.

Appendices

Appendix 1: Proof of Proposition 12.1

Consider first the last stage, that is, stage 3. Remembering that $V^{i}(3,x) =[A_{3}^{i}x + C_{3}^{i}]$ from Proposition 12.1 and Vⁱ(4,x)=0, the conditions in (12.25) become

(12.38)

Performing the indicated maximization in (12.38) yields

(12.39)

The game equilibrium strategies in stage 3 can then be expressed as

(12.40)

Substituting (12.40) into (12.38) yields

(12.41)

Using (12.41), we obtain

(12.42)

Now we proceed to stage 2. The conditions in (12.25) become

(12.43)

Invoking (12.41), the condition in (12.43) can be expressed as

(12.44)

Performing the indicated maximization in (12.44) yields

(12.45)

The game equilibrium strategies in stage 2 can then be expressed as

(12.46)

Substituting (12.46) into (12.44) yields

(12.47)

Collecting the terms together, (12.47) can be expressed as

(12.48)

Substituting $A_{3}^{i} =( \frac{1}{1 + r} )^{2} \frac{P^{2}}{4c_{i}}$ for i∈{1,2} into (12.48), $A_{2}^{i}$ and $C_{2}^{i}$ for i∈{1,2} are obtained in explicit terms.

Finally, we proceed to the first stage. The conditions in (12.25) become

(12.49)

Invoking (12.48), the condition in (12.49) can be expressed as

(12.50)

Performing the indicated maximization in (12.50) yields

(12.51)

The game equilibrium strategies in stage 1 can then be expressed as

(12.52)

Substituting (12.52) into (12.50) yields

(12.53)

for i,j∈{1,2} and i≠j.

Collecting the terms together, (12.53) can be expressed as

(12.54)

Substituting the explicit terms for $A_{2}^{i},A_{2}^{j}, C_{2}^{i}$, and $C_{2}^{j}$ from (12.48) into (12.54), $A_{1}^{i}$ and $C_{1}^{i}$ for i∈{1,2} are obtained in explicit terms.

Appendix 2: Proof of Proposition 12.2

Consider first the last stage, that is, stage 3. Remembering that W(3,x)=[A₃x+C₃] from Proposition 12.2 and W(4,x)=0. The conditions in (12.31) become

(12.55)

Performing the indicated maximization in (12.38) yields

(12.56)

The optimal cooperative strategies in stage 3 can then be expressed as

(12.57)

Substituting (12.57) into (12.55) yields

$$[A_{3}x + C_{3}]= \biggl( \frac{1}{1 + r}\biggr)^{2}\sum_{j = 1}^{2}\frac{P^{2}}{4c_{j}}x,\quad \mbox{for }i \in\{ 1,2\}. $$

(12.58)

Using (12.58), we obtain

$$W(3,x) =[A_{3}x + C_{3}],\quad \mbox{where }A_{3} =\biggl( \frac{1}{1 + r} \biggr)^{2}\sum _{j= 1}^{2} \frac{P^{2}}{4c_{j}} \mbox{ and }C_{3} = 0. $$

(12.59)

Now we proceed to stage 2. The conditions in (12.31) become

(12.60)

Invoking (12.58), the condition in (12.60) can be expressed as

(12.61)

Performing the indicated maximization in (12.61) yields

$$\biggl( P - \frac{2c_{i}u_{2}^{i}}{x} \biggr) \biggl( \frac{1}{1 + r} \biggr)-A_{3}= 0,\quad \mbox{for }i \in\{ 1,2\}. $$

(12.62)

The optimal cooperative strategies in stage 2 can then be expressed as

$$\psi_{2}^{i}(x) =\bigl[P - (1 + r)A_{3}\bigr]\frac{x}{2c_i},\quad \mbox{for }i \in\{ 1,2\}. $$

(12.63)

Substituting (12.63) into (12.61) yields

(12.64)

Collecting the terms together, (12.64) can be expressed as

(12.65)

Substituting $A_{3} =( \frac{1}{1 + r} )^{2}\sum_{j = 1}^{2}\frac{P^{2}}{4c_{j}}$ into (12.65), A₂ and C₂ are obtained in explicit terms.

Finally, we proceed to the first stage, the conditions in (12.31) become

(12.66)

Invoking (12.65), the condition in (12.66) can be expressed as

(12.67)

Performing the indicated maximization in (12.67) yields

$$\biggl( P - \frac{2c_{i}u_{1}^{i}}{x} \biggr)- A_{2} = 0,\quad \mbox{for }i\in\{ 1,2\}. $$

(12.68)

The optimal cooperative strategies in stage 1 can then be expressed as

$$\psi_{1}^{i}(x) =(P - A_{2}) \frac{x}{2c_i},\quad \mbox{for }i \in\{ 1,2\}. $$

(12.69)

Substituting (12.69) into (12.67) yields

(12.70)

Collecting the terms together, (12.70) can be expressed as

(12.71)

Substituting the explicit terms for A₂ and C₂ from (12.65) into (12.71), A₁ and C₁ are obtained in explicit terms.