Negative Beliefs and Suboptimal Outcomes
Game theory teaches us that irrational beliefs can precipitate the very outcomes we fear.
In Econ 35, Prof. Christopher Snyder and Joel Watson1 have introduced me and my peers to the concept of beliefs about other players’ strategies, denoted by θ. We have been taught how to solve for the Player 1’s expected utility provided that Player 1 believes Player 2 chooses action x, y, and z with probabilities a, b, and c, such that a + b + c = 1. Given Player 1’s beliefs, θ2, about Player 2’s mixed strategy, σ2, determining the outcome of the game is trivial enough.
We have not yet discussed how Player 1 forms his beliefs about Player 2’s strategy. As game theory is present all around us, all the time, in ways both big and small, the thought has occurred to me: what if σ2 is a function of what Player 2 believes or knows about Player 1’s beliefs? In short, what if σ2(θ2)?
Let’s assume that Player 1 is pessimistic, bearish, anxious, [insert other detrimental belief-forming attitude here], and his negative belief about Player 2’s strategy, θ2, is such that U1(σ2(θ2)) =< U1(σ2). In English, there are some games in which pessimism about the other player’s strategy brings about the very outcome—or an even worse outcome—dreaded by the first Player. In short, we must be careful when playing some games that our beliefs do not set into a motion a painful self-fulfilling prophecy.
I don’t know about you, dear reader, but I can certainly think of times where my θ2 resulted in an outcome much worse than if I had made no assumptions about the other player’s strategy. Ironically, if I had instead stuck with my original strategy, s1/σ1, and had not made irrational, pessimistic assumptions, θ2, then I would have avoided the outcome I feared and enjoyed the original outcome. In mathematical notation, U1(σ1,σ2) > > > U1(σ1(θ2),σ2(θ2)). What’s worse, U2(σ1,σ2) > > > U2(σ1(θ2),σ2(θ2)). In short, it is wise to remember not to let your beliefs lead you and the partner in your game to a Pareto-dominated, suboptimal equilibrium.
Author of Strategy: An Introduction to Game Theory, 3rd ed. (2013)