Game Theory & Business Strategy

Recommended Book Problems

With Solutions






Problems


Recommended book problems:

Chapter 8 Problem 5 (1st edition)
Chapter 11 Problem 8 (2nd edition)







Solutions


8.5

Kid's Corner
HighLow
Child's PlayHigh64 , 6420 , 72
Low72 , 2057 , 57

(a) To verify that the game is a prisoner's dilemma, we must check that each player has a dominant strategy (low) and that the equilibrium results in lower payoffs for each player than another outcome (64 > 57). Alternately, note that:
High payoff from cheating(72)
> cooperative payoff(64)
> defect payoff (57)
> low payoff from cooperating (20)


(b) Total profits at the end of 4 years = 4 × 57 = 228. If the game is to be repeated a fixed and known length of time, unraveling guarantees that the players will be at equilibrium in each period. Firms know the game ends in 4 years, so they start at the end and use backward induction to find it's best to cheat in year 4 and in each preceding year. No cooperation is sustainable in the finite game.

(c) By cheating, a firm earns 72, but will earn only 57 in every subsequent period. By not cheating, the firm can earn 64 in every period. In order for collusion to be sustainable, we must have
PV(collude) > PV(cheat)
==> 64 + 64/r > 72 + 57/r
==> 7/r > 8
==> 7 > 8r
==> r < 7/8
Thus, if r < 7/8, the grim trigger strategy supports collusion, and if r > 7/8, the firms have an incentive to cheat. If r=0.25, cooperation can be sustained.

(d) Since both firms will be cooperating, each will earn 64 in every period, earning a total of 256 in four years. This is different from part b. since each firm's belief that the interaction is not finite leads to cooperation, whereas in part (b), each firm's knowledge that the game is finite leads to unraveling.

(e) The 10% probability of failure in any given year means that there is only a p=90% chance of seeing the future. Hence, to have a return of $1 a year from now, I must invest not 1/(1+r) today but .9/(1+r). So, if r=0.25, the effective rate of interest becomes:
R = (1 + r - p) / p
= (1 + 0.25 - 0.9) / 0.9
= 0.35 / 0.9
= 0.39 = 39%
This would need to exceed 7/8 for cheating to be worthwhile (see part d), so cooperation would still be sustained.
If the probability of bankruptcy rises to 35%, p=0.65, and the future becomes worth less. Then, R=92%. So, if bankruptcy becomes more certain, cheating becomes worthwhile.