The cooperative strategy is defined as the best joint outcome for both players together.
Cooperative Strategy = A strategy that leads to the highest joint payoff for all players.
Thus, the cooperative strategy is identical to collusion, where players work together to achieve the best joint outcome. In the Prisoner’s Dilemma (Figure 6.1), the cooperative outcome is found by summing the two players’ outcomes together, and finding the outcome that has the smallest jail sentence for the prisoners together: \(\text{(NOT, NOT) } = (3, 3)\).
This outcome is the collusive solution, which provides the best outcome if the prisoners could make a joint decision and stick with it. Of course, there is always the temptation to cheat on the agreement, where each player does better for themselves, at the expense of the other prisoner.
Similarly, the cooperative outcome in the advertising game (Figure 6.2) is \(\text{(AD, AD) } = (20, 20)\). This outcome provides the highest profits \((= 40\) million USD) to both firms. Note that the advertising game is not a prisoner’s dilemma, since there is no incentive to cheat once the cooperative solution has been achieved.
Game Theory Example: Steak Pricing Game
A pricing game for steaks if shown in Figure \(\PageIndex{1}\). In this game, two beef processors, Tyson and JBS, are determining what price to charge for steaks. Suppose that these two firms are the major players in this steak market, and the outcomes depend on the strategies of both firms, since players choose which company to purchase from based on price. If both firms choose low prices, the outcome is low profits. Additional profits are earned by choosing high prices. However, when both firms have high prices, there is an incentive to undercut the other firm with a low price, to increase profits at the expense of the other firm.
Figure \(\PageIndex{1}\): Steak Pricing Game: Two Beef Firms. Outcomes in million USD.
Steak Pricing Game: Dominant Strategy
(1) If \(\text{TYSON LOW, JBS}\) should \(\text{LOW } (2 > 0)\)
(2) If \(\text{TYSON HIGH, JBS}\) should \(\text{LOW } (12 > 10)\)
…the dominant strategy for \(\text{TYSON}\) is \(\text{LOW}\).
(3) If \(\text{JBS LOW, TYSON}\) should \(\text{LOW } (2 > 0)\)
(4) If \(\text{JBS HIGH, TYSON}\) should \(\text{LOW } (12 > 10)\)
… the dominant strategy for \(\text{JBS}\) is \(\text{LOW}\).
The Equilibrium in Dominant Strategies for the Steak Pricing game is \(\text{(LOW, LOW)}\). This is an unexpected result, since it is a less desirable scenario than \(\text{(HIGH, HIGH)}\) for both firms. We have seen that an Equilibrium in Dominant Strategies is also a Nash Equilibrium and a Minimax Equilibrium. These results will be checked in what follows.
Steak Pricing Game: Nash Equilibrium
(1) Outcome \(= \text{(LOW, LOW)}\)
(a) Is \(\text{LOW}\) best for \(\text{JBS}\) given \(\text{TYSON LOW}\)? Yes.
(b) Is \(\text{LOW}\) best for \(\text{TYSON}\) given \(\text{JBS LOW}\)? Yes.
…\(\text{(LOW, LOW)}\) is a Nash Equilibrium.
(2) Outcome \(= \text{(LOW, HIGH)}\)
(a) Is \(\text{LOW}\) best for \(\text{JBS}\) given \(\text{TYSON HIGH}\)? Yes.
(b) Is \(\text{HIGH}\) best for \(\text{TYSON}\) given \(\text{JBS LOW}\)? No.
…\(\text{(LOW, HIGH)}\) is not a Nash Equilibrium.
(3) Outcome \(= \text{(HIGH, LOW)}\)
(a) Is \(\text{HIGH}\) best for \(\text{JBS}\) given \(\text{TYSON LOW}\)? No.
(b) Is \(\text{LOW}\) best for \(\text{TYSON}\) given \(\text{JBS HIGH}\)? Yes.
…\(\text{(HIGH, LOW)}\) is not a Nash Equilibrium.
(4) Outcome \(= \text{(HIGH, HIGH)}\)
(a) Is \(\text{HIGH}\) best for \(\text{JBS}\) given \(\text{TYSON HIGH}\)? No.
(b) Is \(\text{HIGH}\) best for \(\text{TYSON}\) given \(\text{JBS HIGH}\)? No.
…\(\text{(HIGH, HIGH)}\) is not a Nash Equilibrium.
Therefore, there is only one Nash Equilibrium in the Steak Pricing game: \(\text{(LOW, LOW)}\).
…\(\text{JBS}\)’ Maximin Strategy is \(\text{LOW } (0 < 2)\).
(2) \(\text{TYSON}\)
(a) If \(\text{LOW}\), worst payoff \(= 2\).
(b) If \(\text{HIGH}\), worst payoff \(= 0\).
…\(\text{TYSON}\)’s Maximin Strategy is \(\text{LOW } (0 < 2)\).
The Maximin Equilibrium in the Steak Pricing game is \(\text{(LOW, LOW)}\). Interestingly, if both firms cooperated, they could achieve much higher profits.
Both JBS and Tyson can see that if they were to cooperate, either explicitly or implicitly, profits would increase significantly. The cooperative outcome is \(\text{(HIGH, HIGH) } = (10,10)\). This is the outcome with the highest combined profits. Both firms are better off in this outcome, but each firm has an incentive to cheat on the agreement to increase profits from 10 m USD to 12 m USD.
