economics · maths

Cournot healthcare

There is an interesting branch of mathematics called Game Theory. It deals with rational decision making in the face of strategic possibilities. One of the most famous examples of such a game is the prisoner’s dilemma, in which two prisoners, A and B, get the choice of keeping their mouth shut or betraying the other, presuming that they have some dirt on each other. The strategic options are the following:

  • If A and B betray each other, they both serve 2 years in jail.
  • If A betrays B but B keeps his mouth shut, A will go free but B will serve 3 years in jail. The same holds if we swap A and B.
  • If both keep their mouth shut, both will serve only 1 year in jail.

The prisoners are not allowed to communicate about their choices. Clearly, the best outcome from an outsider’s perspective will be if both keep their mouths shut, so that both serve only one year in jail. If they were able to talk, they might reach that conclusion too. However, since they can’t talk, they will have to consider the possibility that the other betrays them. This is a simple two-choice strategic game.

As it turns out, it’s always better to betray the other, as we may see by considering the options of the opponent. Let’s take A’s perspective for a moment. If B betrays us, we’re better off betraying him too, since that’ll save an extra year in jail. However, if B doesn’t betray us, we’re also better off betraying him, since then we go scot-free. Supposing that we are purely self-interested, our best response in either case is to simply betray B. However, B will probably come to the same conclusion, so that both prisoners end up playing their best responses to each other.

This solution concept to a game is called a Nash equilibrium: the set of strategies where all players are playing their best responses to everyone else. Such analysis also lends itself to model two competing firms producing the same commodity. One famous example is called Cournot competition. The Nash equilibrium of the Cournot game is the natural outcome if both firms know how much the other firm is producing and choose their best response. It turns out that the Nash equilibrium leads to prices that are somewhere in between monopoly and perfect competition prices (selling at cost). In a two-firm market, consumers end up paying too much even when the two firms aren’t allowed to collude. However, as the number of firms increases, the equilibrium price converges to the cost.

It seemed to me to be an interesting little model to explore healthcare insurance firms. Healthcare insurance firms, however, don’t produce anything. In essence, they serve as a big pot to redistribute money to people who actually need it. We all pay a little bit and when someone falls ill, they can get treatment from the communal fund. Ideally, everyone ends up paying less than if everyone would only take care of themselves. How much someone pays, the premium, is ideally calculated so that the whole system breaks even on average, based on statistical data. In reality, it is of course possible that, like in Cournot competition, consumers end up paying more than in the ideal case.

To set up the Cournot healthcare game, lets think about how a healthcare insurance firm works. Everyone pays a fixed amount of money, the premium, while the firm is in charge of redistributing that money. The cost for each firm we can take as the total amount of money that they give back to their customers. Of course, they also have to pay their employees, building rent etc., but we can include this in the premium. Let assume for simplicity that both firms charge the same premium, and their method of competing is by adjusting the quality of their service. Quality we can simply define as the ratio of cost and total revenue. In the ideal case that a firm spends all the money on customers who need healthcare, the quality measure is one, while the quality measure is zero if they keep everything.

To put it into a set of equations, we have the profit, \Pi_i, of firm i

\Pi_i = a n_i (1-q_i),

with a the premium, n_i the number of customers and q_i the quality measure. Naturally, the total number of people is fixed, so we have the constraint that n_1 + n_2 = n. Furthermore, in Cournot competition the price is determined by a linear relation of the total production. In our case, the number of people that choose one firm rather than the other should therefore depend linearly on the quality. So the number of customers should satisfy

\frac{n_1}{n_2} = \frac{q_1}{q_2}.

One could argue that the firm that provides the highest quality should get all the customers, but I don’t think this is true. The customers, of course, don’t actually know the quality of the firms. Furthermore, the quality is a statistical measure, one that your average customer probably has no access to. However, it is likely that customers that are denied service will switch firm, leading to something like the previous relation. The number of customers that choose firm i can be shown to be

n_i = n \frac{q_i}{q_1+q_2}

The profit for firm i therefore satisfies

\Pi_i = a n \frac{q_i}{q_1+q_2}(1-q_i).

For simplicity’s sake, I will switch to the relative profit margin, being p_i = \frac{\Pi_i}{a n}, so that our model actually becomes independent of the number of people or the premium.

In free market competition, the best response of firm 1 to firm 2 can be found by maximising profit. If the market is not free and there are, for instance, legal constraints to what healthcare should be provided, the situation is different. Lets assume for the moment that competition is free, so that a firm will set its quality such that profit is maximised. This can be done by differentiating the profit with respect to the quality, and setting the result to zero. For firm 1 this means

\frac{\partial p_1}{\partial q_1} = -\frac{q_1^2 + 2q_1 q_2 - q_2}{(q_1+q_2)^2} = 0.

The denominator is clearly positive unless both firms are only cashing in and not providing any healthcare whatsoever. However, in this case it seems unlikely that anyone would get healthcare insurance in the first place, so we may ignore this possibility. Therefore, we are led to a quadratic equation for q_1, reading

q_1^2 + 2q_1 q_2 - q_2 = 0.

Recalling our high-school maths, we find that

q_1 = \sqrt{q_2^2 + q_2} - q_2.

There is a negative root to this equation as well, but we assumed that the quality has to be between zero and one. This is the best response, in terms of quality, for firm 1 provided they know what the quality of firm 2 is. We can repeat this exercise for firm 2, but since the game is symmetrical, we can simply swap indices. These two response curves can be plot, and they look as follows.


There are several interesting things to note about this figure. First, there are two intersections, one at (q_1,q_2) = (0,0) and another one that is nontrivial. These are the Nash equilibria, situations where each firm is giving the best response to the other. Whenever the qualities are equal, such as in an equilibrium, the population is exactly divided in half over the two firms. We can also see that the maximal quality in the best response curve is quite low. If firm 2 is providing 100% service to their customers, firm 1 should apparently only respond with a quality of q_1 = \sqrt{2}-1, which is roughly 41%, severely limiting healthcare to people who need it.

The trivial Nash equilibrium, (q_1,q_2) = (0,0) happens to be unstable. If firm 2 uses a very small but nonzero quality, the best response of firm 1 is to provide a much larger quality. For instance, if firm 2 decides to use a quality of q_2 = \frac{1}{15}, the best response for firm 1 will be to use a quality of q_1 = \frac{1}{5}, which is a factor of 3 higher. Firm 2 may then respond to the new situation, increasing its quality over the quality of firm 1. Fairly quickly, both firms will converge on the nontrivial Nash equilibrium.

Let us now find this situation, the Nash equilibrium where both firms have nonzero quality while giving a best response to each other. This happens where the curves intersect, so that both best response qualities are equal,

q = \sqrt{q^2+q}-q,

Some algebra will reveal two solutions, one being the trivial equilibrium q = 0, the other being q = \frac{1}{3}. From a societal point of view, this is hardly a desirable outcome. We would of course want, perhaps even demand, that any healthcare insurance firm will provide full service. However, as long as those firms also have to make a profit, as they will aim to do in a free-market situation, they will not fully expend all the necessary healthcare.
In fact, one can do the whole analysis for any number of firms, say N, and the result is that the equilibrium quality will be

q = \frac{N-1}{2N-1}.

It is easy to see that the quality will go to \frac{1}{2} in the limit of an infinite number of firms, N \to \infty. For any finite number of firms, the equilibrium quality will always be below a half. Roughly speaking, people will never receive more than half the healthcare they actually need.

So what now? What should we do to increase the quality of healthcare insurance above 50%? The answer lies in the free market assumption. We may create and enforce laws that insurance firms will have to satisfy. If we enforce some minimal quality, for instance, then any firm will simply provide the legally minimal quality.
Suppose, however, we don’t assume the best response is defined by maximising profit. In effect, we would be turning to a different metric. If we optimise for quality, let’s say, we trivially end up with 100% service. The take home message is this: free market competition in healthcare insurance leads to people receiving less healthcare than they actually need. Instead of optimising for profit, we should optimise for the number of people that actually get healthcare.

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