The Cheating Equilibrium
Suppose some agent i has to produce a specific task. The output is measured in terms of a performance P. The minimum value of P is 0 and can be either bounded from above or not. P depends on two variables: a random and normally distributed variable r which measures the agent’s (non-chosen) talent and luck in executing the task and a variable c measuring the extent i is cheating to increase her performance. Cheating consists here in using non-allowed means to execute the task, e.g., using tools or performance-enhancing products such as drugs. Another agent j observes P, but knows the value of neither r nor c, though we assume that he knows the distribution of r. While i’s utility is monotonically increasing with P (i.e., i wants to perform as best as possible), j’s sole interest is to determine whether or not i has cheated. Hence, j’s preferred outcome is to guess right and if he is wrong, we assume that he prefers to not detect a cheating agent i than to accuse a non-cheating agent i. Agent i must choose the level of cheating such as to maximize P without being caught cheating. If she is caught, her utility u(P) = 0; if she is not caught, u is monotonically increasing with P (if i is wrongly accused of cheating, we assume for the moment that this is known, and u is then a function of P as when she is not accused).
In this setup, because j knows the distribution of the values of r, he can infer (provided that he also knows the “production function”) the expected value of P if no cheating occurs. Denote this expected value Pnc*. If P < Pnc*, it is basically impossible for j to infer whether or not cheating has occurred, and absent any other information, the second-best is to never accuse. What about P > Pnc*? Obviously, j cannot be sure that i has cheated, but (because r is normally distributed) the more P is above Pnc*, the higher the probability that cheating has occurred. At the extreme, in case P is bounded from above, the probability of a maximal performance is almost null; it will always almost pay off for j to accuse i. More generally, there is a value P* such that it is j’s best response (in expectation) to systematically accuse i. Hence, i will never set c at the maximum, or at any level such that (given r) her performance P is above P*. The cheating equilibrium c* corresponds to the situation such that maxc*ui(P(r, c*)) such that P(r, c*) < P*. In the general case, i is thus cheating to some extent at the equilibrium.
We can complicate this basic model in different ways. First, we can assume that i must support a monotonously increasing moral cost if she chooses to cheat. If this moral cost is positive and deterministically known, it decreases the equilibrium value c*. But we can assume that depending on her type, agent i may incur a more or less important moral cost for a given level of cheating, and that j ignores i’s type. In this case, depending on the distribution of types, this informational asymmetry may increase the level of c*. Second, we may consider that the effect of c on P is either deterministic or stochastic. In the latter case, depending on i’s risk attitude, this will affect the value of c* (positively if i is risk-prone, negatively if she is risk-averse). Again, j may ignore i’s risk-attitude. Third, we may assume that j can choose to engage in costly monitoring actions to detect cheating. Such actions increase the probability of detecting cheating when it occurs by giving more information (technically, the more is invested in these actions, the higher the probability for j to non-ambiguously detect cheating if it occurs). The effect would be to lower the value of c*, while the latter would still be positive in the most general case. Finally, we could assume that whether or not i has cheated always remains ambiguous, even after the game has ended. The utility gains of i and j should then be discounted accordingly. The effect on the equilibrium is not clear. Everything else equals, it makes less valuable for j to accuse i of cheating because in the end, uncertainty will remain anyway about what has really happened. It will therefore raise P* and so c*. But correspondingly, it becomes less risky to falsely accuse someone because doubts will remain, lowering both P* and so c*. On the other hand, because now i can be falsely accused of cheating without being able to definitely prove her innocence (contrary to what I have assumed at the beginning), she has an additional interest in cheating anyways. But because j may be prone now to accuse i even for a relatively low performance level, it encourages i to reduce her level of cheating.
I think this generic model and its extensions capture well the point Tyler Cowen makes in this essay, in relation in particular to the recent affair in the chess world created by the forfeit of world champion Magnus Carlsen in an important tournament following a loss against an opponent that, some (Carlsen included, though he has not explicitly formulated the accusation) believe, has used illegal means in this game as well as in other instances:
A subtler truth is that a lot of the cheating will be modest and marginal rather than blatant. Consider computer cheating in chess. If you find a way to consult the computer every move, you will win every game with near-perfect play. You will also be caught immediately. So you might cheat for only a few moves every game — enough to help but not so much to be detected. Given that both sides will employ countermeasures, and detect suspicious instances of clearly superior performance, a lot of cheating will be pretty mediocre, and deliberately so.
As decisive moments approach, games and competitions might become less honest — and tensions in the crowd will rise as people wonder whether they are watching the real thing or some AI-aided simulacrum. Brilliancies will forever be called into question. Dishonest players, in turn, will have to carefully consider when to exercise their de facto “cheating privileges.”
I think the last two sentences are especially important: as cheating methods because more sophisticated and make cheating more effective and difficult to detect, talent and performance will always raise suspicion. It’s sad news for authentically talented people. The good news is that because suspicion is raising, cheaters will have to content themselves with a low level of cheating, with only a mediocre impact on their performance. In other words, the world of sophisticated cheating techniques will be a world of endemic low-level cheating where everyone except for the most talented cheats without being able on average to achieve the highest levels of performance of the truly talented people.