# Passing the Plate for Universal Health Insurance?

My Regular Libertarian Foil ™ (RLF – an acquaintance who often provokes me with unorthodox points of view) recently posed the following question: Why can’t all the liberals who think universal health insurance is a moral issue take up a collection and fund it themselves without raising my taxes?

This question put me in mind of a problem I encountered in Ben Polak’s game theory course.  Suppose that a person can make an investment of ten dollars that will return fifteen dollars (net return of five dollars), only if ninety percent of everybody also invests.  If less than ninety percent invest, an individual’s return on his or her ten dollar investment is nothing.  When Professor Polak first posed this problem to his class of Yale students, only about half chose to invest, and they all lost their investments.  A second round of the game produced very few investors, and a third round almost none.  He then explained the dynamics of the game and its payoffs.  If I believe that nearly everybody will invest, then my best response to their expected play is to also invest, and the same is true for everybody else who shares the same belief.  In this case, the solution set of everybody investing is said to be a “Nash equilibrium,” because nobody can do better by not investing given their belief that others will as well.

But there is another Nash equilibrium in this game.  If I believe that not enough people will invest, then my best response to others’ expected play is not to invest either, and the same is again true for everybody else who shares that belief.  We saw actual play converge on this Nash equilibrium as the members of the class set their expectations in response to others’ play in previous rounds.  This was unfortunate, however, because the payoffs associated with the Nash equilibrium of everybody investing are superior for everybody (technically, “Pareto superior“).  Happily, solving the coordination problem that frustrated the superior equilibrium in the initial play was as simple as explaining the game, as everybody chose to invest during a final round of play at the end of the lesson.

It is simple to construct a similar model for funding universal health insurance through a voluntary tax on liberals.  Suppose that a voluntary tax on each liberal in the amount of X would be sufficient to fund universal health insurance, provided that ninety percent of liberals paid the tax.  Suppose also that each liberal experiences an increase in utility (happiness, moral satisfaction) equivalent to (i.e, for which they would have willingly paid) 2X when there is universal health insurance.  Clearly, each liberal is better off when all liberals choose to pay the tax than if no liberals do.  But is there a Nash equilibrium in which all liberals pay the tax?  There is not.  If I pay the tax and all the others do too, my expected payoff is X — the 2X utility from having universal health insurance less the X I paid in tax.  But if all the other liberals pay and I don’t, my payoff is 2X.  So not paying is my best response to the others’ expected play if I believe that they will pay the tax.  And of course, my best response to others’ expected play if I believe a sufficient number of them will not pay the tax is also not to pay the tax.  The game has one Nash equilibrium in which no liberal pays the tax and there is no universal health insurance, even though liberals would be collectively better off if everybody paid the tax.  It is, in short, a prisoner’s dilemma.

One potential solution to a prisoner’s dilemma is cooperation.  We can all agree to pay the tax for our mutual benefit.  But of course, each of us has an incentive to cheat and receive the benefit without paying a share of the cost.  And studies of successive-round prisoner’s dilemma type games have shown that even players who are initially inclined to cooperate will revert to their equilibrium strategies as they perceive or fear that others are cheating.  So the expected outcome even if liberals all agree to pay the tax is eventually the same without some enforcement mechanism to prevent cheating.  We could make the liberal tax mandatory, but then we would have to identify the liberals.  Though I’m sure my RLF would rejoin that he finds them easy enough to spot, you get the picture.  Private charity, unsurprisingly, can’t solve the problem of universal health insurance.