Game Theory: An Introductory Sketch

Nash Equilibrium and the Economics of Patents

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Game Theory: A Non-Technical Introduction to the Analysis of Strategy

In the Queueing Game, the first person in line gets the best service. In a patent system, the first person to invent a device gets the patent. Let's apply that parallel and try to come up with a game-theoretic analysis of patenting. For this example, we will have to try to think a bit like economists, and use two economic concepts: the concept of diminishing returns to investment and the focus on the additional revenue as a result of one more unit of investment -- the "marginal revenue," in economist jargon. If you have studied those concepts, this will be your reminder. If you haven't studied them, take it slow, but don't worry -- it will all be explained.

Patents exist because inventions are easy to imitate. An inventor could spend all his wealth working on an invention, and once it is proven, other businessmen could imitate it and get the benefit of all that investment without making any comparable investment of their own. Because of this, if there were no patents, there would be little incentive to develop new inventions, and thus very few inventions. At least, that's the idea behind the patent system. But some economists point out that patents are limited in time and in scope, so that there probably isn't as much incentive to invent as we would need to get an "efficient" number of inventions. This suggests that we don't get enough inventions -- but that may be a hasty conclusion.

For this game example, let us think of some new invention. At the beginning of the game, a number of development labs are considering whether to invest in the development of the invention. It is not known whether the invention is actually possible or not. Research and development are required even to discover that. But everyone estimates that, if the invention is produced and patented, it will yield a profit of $10,000,000. To keep things simple, we ignore any other potential benefits and proceed as if the profits were the only net benefits of developing the invention. Also for simplicity, we suppose that the only decision a laboratory can make is to spend $1,000,000 on a development effort or to spend nothing. A lab is not allowed to invest more nor any positive amount less than $1,000,000.

If there are investments in development, the invention may or may not be successfully developed. Since we don't know whether the invention is possible or not, the development effort may fail, and we can only say how probable it is that the invention will be made. The probability that the invention is successfully developed depends on the amount invested by the whole group, as shown in Table 2. But what does this probability mean, in money? To answer that question, we compute the "expected value" of total revenue -- that is, the revenue of $10,000,000 if the invention is made times the probability. Thus, if the probability of success is fifty percent, the "expected revenue" is 0.5*$10,000,000 = $5,000,000. The more labs invest, the greater the probability of success is, up to a point. But the labs' investment is subject to "diminishing returns:" at each step, the additional $1,000,000 of investment increases the expected revenue by a smaller amount. This is shown in the last column, as the "additional expected revenue" goes from $3,000,000 to $2,000,000, and so on.

Table 2

How much should be invested, for maximum net benefits? Economic theory tells us that it makes sense to keep increasing the investment as long as the additional expected revenue (marginal benefit, in economist's jargon) is greater than the investment necessary (marginal cost, in economist's jargon). That means it is efficient to increase investment in development as long as adding one more lab will increase the expected revenues by at the amount the lab will invest, $1,000,000, but no further. The second lab adds 2 million, while the third adds only 667,000 in expected revenues. The third lab is a loser, and the efficient number of labs working on developing this invention is two.

But how are these payoffs distributed among the development labs? We assume they are distributed on the "horse-race" principle: only the lab that "comes in first" is a winner. That is, no matter how many labs invest, 100% of the profits go to one lab, the lab that completes a working prototype first. Since all of the labs invest the same lump sum $1,000,000, we shall assume that all who invest have an equal chance of getting the payoff, if there is any payoff at all. Thus, when two labs invest, each has a 50% chance at a 50% chance of a $10,000,000 payoff -- that is, overall, a 25% (50% times 50%) chance at the $10,000,000, for an expected value payoff of $2,500,000 and a profit of $1,500,000. How many will invest in this "horse race" model of invention? If an enterprise is considering investing when two others are already committed to investing, it can anticipate gaining a 1/3 chance at a 56.67% chance (overall, an 18.89% chance) at $10,000,000, for an expected value of $1,889,000 and a profit of $889,000. The third lab will invest. What about the fourth, fifth, sixth? To make a long story a little shorter, the sixth lab would gain a 1/6 chance at a 61% chance at $10,000,000, for an expected value of $10,166,666.67 and a profit of $166,666.67. The sixth (fourth, and fifth) labs will invest. However, a potential seventh lab will anticipate a 1/7 chance at a 61% chance at the $10,000,000 payoff. This is an expected value of $871,428.57 and a loss of $128,571.43. The seventh firm will not invest. Any set of strategies in which six firms invest in research to produce this invention is a Nash-equilibrium.

Thus, in equilibrium, six firms will contest this particular "horse race," and that is three times the efficient number of labs to work on developing this invention. The allocation of four more labs to this job increases the probability of success by just 11%, a change worth just over a million; but four million are spent doing it!

Notice that this model could be applied when the reward goes, not to the first innovation in time, but to the first on some other scale. For example, suppose that the information products developed are not patentable, but are slightly different in design, as different computer applications might be. Suppose also that only the one perceived as "best" can be sold, and the rest fail and investments in developing them are lost. Finally, suppose that enterprises that invest equally have equal chances of producing the "best" dominant product, the "killer app." This "star system" will create a tendency toward overallocation of resources to the production of the information products.

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Roger A. McCain