The previous lecture explained Coasean bargaining between two parties. This lecture is a primer on game theory: the strategies that parties may follow when some of the strong assumptions that support Coasean bargaining (zero transactions costs, full information, perfect compliance, etc.) are violated. What if the hatchery suspects the paper mill is sneaking more effluent into the river than they agreed on, or the effluent turns out to be more toxic than the hatchery anticipated? What if the mill thinks the hatchery is planning another lawsuit?

The classic formulation of the single-iteration two-party strategy game is known as the "prisoners' dilemma." Suppose the police have arrested two burglars after a break-in and are interrogating them separately.  If they can both cooperate and keep silent, they will both go free.  But if either suspect confesses and helps convict the other, he will get 2 years in prison while his accomplice will get 10 years.  If they both confess, they each get 6 years.  The optimal outcome would be for both suspects to hold out, but since neither suspect can be sure the other isn't ratting him out, there is a strong incentive to confess, so the likeliest outcome is for both of them to plead guilty and serve 6 years each.   The strategy game can be represented in a payoff matrix.  Here the cell values represent prison terms for A and B respectively:

Confessing is a "max-min" strategy: in case the other party chooses the strategy that includes the worst possible outcome for you, you should choose the strategy that yields the best of these bad outcomes for you.  You make the best of the worst-case scenario.

The ultimate prisoners' dilemma during the Cold War was:

The risk of a pre-emptive strike fell as the two nations designed systems to insure their weapons could survive a first strike (hardened ground-based missile silos, launches from subs, etc.) and retaliate with total destruction. Nobody would win. The growing certainty of the MAD ("Mutually Assured Destruction") scenario reduced the incentives for a first strike, and restrained both nations to the mutual stand-off scenario.

Some games may have altruistic solutions, where the players cooperate to maximize their combined benefit. Suppose a couple chooses between a football game (his preference) or a skating exhibition (her preference) when it's not much fun being apart.  The payoffs are his and her utilities:

In general, where cooperation is not enforceable, parties can be expected to cheat any time they have sufficient incentive to defect from the cooperative arrangement.  Let $C be the reward for Cooperation; $T be the Temptation to defect; $S be the Sucker's reward for cooperating and getting betrayed; and $N be the payoff for Non-Cooperation:

A Nash equilibrium is when, once the players have chosen their strategies, no player can benefit by changing his or her own strategy when the other players keep theirs unchanged.  In general, the non-optimal solution $N,$N will prevail when $T > $C > $N > $S.  Either player will defect in response to the other's defection. It is a Nash equilibrium if $N > $S.

Games can be expanded in various dimensions: more players (8 nations are now in the "Nuclear Club" and we'll probably have non-state players in the future), more strategy options per player, differential payoffs (e.g., $C_A < $C_B); multiple iterations of the game from which players can learn from each others' strategies, etc. 

Game theory explains the behaviors of oligopolists who tacitly agree not to compete on price while maintaining a charade of price competition, like the same discounts on Coke and Pepsi in alternate weeks at the supermarket. As the number of firms in the oligopoly increases, enforcing cooperation becomes more difficult, and the oligopoly loses market power.

OPEC's production quota negotiations are a high-stakes oligopoly game. if all members adhere to their agreed-on quotas, oil prices remain high and they all benefit.  But the high prices give each OPEC member a strong incentive to sneak extra oil into the market.  This increases short-run profits for the cheaters, but reduces oil prices and long-term profits for everyone. 

A number of economists have conducted iterative game experiments to determine how variations in the payoff matrix affect the cooperation and retaliation strategies of experimental subjects (typically pairs or groups of student volunteers). If the winning iterative-game strategy is to cooperate, subjects typically learn to use defection only for retaliation for another player's defection in the previous round.

Here's a game with no equilibrium:

Mutual cooperation is Pareto-optimal, but B has an incentive to defect, which causes A to defect, which causes B to cooperate, which causes A to cooperate, etc.: the game cycles clockwise over multiple iterations until the players learn that mutual cooperation is optimal and B accepts that his payoff will be less than A's. After the players figure this game out, B might argue that, in the interest of "fairness," he should have the right to defect and collect $4 occasionally. It could be worthwhile for A to bribe B to eschew this strategy.