The Purpose of the Law
Some contend that the purpose of the law, particularly in the commercial field, should be to create incentives and rules such that the strategies people select in the pursuit of self-interest also happen to maximize the total wealth of relevant persons. This Demonstration examines the set of legal rules that will fulfill this purpose. You assume a simple legal setting such as a contract or two-party accident that can be represented as a two-player strategic form game in which each player has two strategies. The strategies are sorted such that the top-left strategy always maximizes total wealth. The idea is to find the set of transfers of wealth such that the top-left strategy is also a pure Nash equilibrium. "Transfer of wealth" means that, if a certain strategy combination is played, the first (row) player transfers this (possibly negative) amount of money to the second (column) player. You control several parameters in this exploration. First, you select from among 32 sample sets of initial payoffs. Second, to permit three-dimensional visualization, you select a strategy combination whose payoffs the law will leave untouched. By default this is the bottom-right strategy combination. Third, you select the amount of "waste" permitted in legal transfers for any of the strategy combinations that the law may affect. "Waste" means that some of the money lost by the first player is not gained by the second player; it is just lost in the transaction. The Demonstration outputs the strategic form of the original game and five sample solutions that make the top-left strategy combination into a pure Nash equilibrium. You can display these solutions by selecting one of two tabs: the first gives the total payoffs, the second gives the transfers of wealth required. The right side of the output displays a polytope that is the part of the set of solutions that will make the top-left strategy combination the Nash equilibrium. (In many instances, the set will actually extend indefinitely outside the plotted region.) The exterior of the set is colored according to the wealth of the first and second players.
In a conventional representation of a two-player strategic form game in which there is a mapping from strategy combinations to payoffs, the first player (the row player) has its payoff shown on the top left; the second player (the column player) has its payoff shown on the bottom right of each cell.
One point of this Demonstration is to show the multiple legal rules that can often create wealth maximization. It thus takes more than a theory of wealth maximization to decide amongst the often infinite number of legal rules. Such decisions might be made on the basis of distributional equality or other concepts of justice.
The solution to this problem depends on the solution of a number of inequalities: (1) the payoff to the row player from the top-left strategy combination has to be higher than the payoff to the row player from the bottom-left strategy; (2) the payoff to the column player from the top-left strategy combination has to be higher than the payoff to the column player from the top-right strategy combination; (3) to assure uniqueness, it cannot be the case that both (a) the payoff to the row player from the bottom-right strategy combination is higher than the payoff to the row player from the top-right strategy combination and (b) the payoff to the column player from the bottom-right strategy combination is higher than the payoff to the column player from the bottom-left strategy combination; and (4) the "waste" resulting from each transfer may never be negative: the players cannot create wealth in the game, only transfer or destroy it. Mathematica solves this combination of inequalities using the command CylindricalDecomposition.
The title "The Purpose of the Law" is said a bit tongue-in-cheek.