By: Ashley Pandya

Game theory, stock markets, and questions best left unanswered

Against my better judgment, I’m a late sleeper.

I have my justifications. It is in those groggy, 4AM, dear-God-won’t-somebody-please-knock-me-out-with-a-baseball-bat moments that I engage in my most potent self-sabotage. I am driven by an unseeable force to inexplicably start asking myself where I fall in the universe: How smart am I, really? Indeed, more troublingly: How smart is everyone else?

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Questions like these are the foundation of microeconomic game theory: a set of theories and models designed to study and predict strategic interactions between economic agents. When game theory is applied to economic markets, its utilization often belies the critical assumption of mutual rationality among players. The doctrine of rationality applied to basic economic theory asserts the following: in a simple strategic interaction between two players – a game of tic-tac-toe, for instance – both of the following must be true for each player in the game for the rational choice model to apply:

(1)   The player must be operating perfectly rationally, optimizing their choices given all information available to them, making decisions using all the intellectual power available to them.

(2)   They must be under the reasonable assumption that their fellow player(s) are also operating  in accordance with (1).

Hm.

In many cases, this isn’t a bad assumption to make. But empirically – in scenarios that are readily conceivable to economically laypeople – the idea that we operate in the most efficient and rational way possible at all times is a little absurd. Even more risible is the notion that we actually believe that everyone else is doing the same. Yet contemporary economists are well aware of this! Models are abstractions. They do not, and indeed do not intend to, accurately reflect the world. Were there to be a model that reflected the world perfectly and in all its detail, there would be no reason for the model to exist at all. While mutual rationality is often a reasonable assumption to make while playing simple strategic games like tic-tac-toe (or common sports like football), traditional game theory needs to be supplanted with more complicated models when participants are left in the dark about the aptitude of their fellow players.

         Such is the conceit of what are known as “Family-Feud”-style experiments: thought puzzle in which participants are asked to make decisions based on their assessment of how their fellow participants will respond. The experiment targets expectations players have about their fellow players’ beliefs, biases, and/or aptitudes. One such experiment is known as the “Guess 2/3 of the average” puzzle. First posited by Alain Ledoux in a 1981 edition of the magazine Jeux et Stratégie, participants are asked to guess what 2/3 of the average guess of all participants’ answer will be. The Nash equilibrium of this game – which we can colloquially read here as the best answer that someone could’ve come up with, given how everyone else responded, under the stipulation that all players are perfectly rational – would be zero.

         Consider how one would arrive here.

         “Level 0 Thinker”: Doesn’t attempt to engage with the mechanisms of the game. Since their choices are all integers from 1-100 (inclusive), they choose the expected value, 50.

         “Level 1 Thinker”: Believes that their participants are, on average, Level 0 thinkers. In less objective terms, they have little faith in their fellow participants. They will choose 2/3 of 50, or 33.

          “Level 2 Thinker”: Believes that their participants are, on average, Level 1 thinkers. Has a little bit more faith in their fellow players than the Level 1 thinkers. They will choose 2/3 of 33, or 22.

          “Level 3 Thinker”: Believes that their participants are, on average, Level 2 thinkers. Has a little bit more faith in their fellow players than the Level 2 thinkers. They will choose 2/3 of 22, or about 15.

         And so on. As players anticipate higher-level thinking on their fellow players’ part, their average guess gets smaller and smaller. When they assume that their fellow players are perfectly rational (the “highest level of thinking”, in some respects), traditional game theory predicts that the number will reach the Nash equilibrium of zero. This analysis can extend itself past ideal rationality into the more realistic pretty-good rationality: that is, even if we don’t imagine that individuals are perfectly rational (meaning that the result of games like these will never truly reach the Nash equilibrium), but we expect that they are generally very rational, the experiment’s results can still hold meaning. The smaller the number, the greater the expectation of rationality.

         What we have come to realize with repeated incarnations of this experiment, however, is that the results say less about the inherent aptitude of participants, and rather about our own assessments of them. Take perhaps the most consequential game board we know of: the financial market. When Richard Thaler replicated the “Guess ⅔ of the Average” experiment in The Financial Times in 1997, the expected result given the strong economic aptitude of the average participant in the experiment was, if not exactly zero, pretty close to it.

         Thaler’s result was 13.

         What’s going on here? Among other things, players are responding to imperfect information they have about the aptitude of their fellow players, and are resorting to personal heuristics that are riddled with biases. When this imperfect information permeates a game space (as it does in almost all natural experiments), players tend to act in contravention with traditional game theory and usually expect some level of irrationality from their fellow players.

        Let’s head to the stock market to see how this plays out in the real world. According to macroeconomic theory, individuals should invest in a firm if expectations of the firm’s future profitability are high. This can be determined by the financial value known as “Tobin’s q”, which represents the ratio of an asset’s market value (as determined by the price of its stock) to its replacement value.

Tobin’s q = Market Value / Total Replacement Value

If the price of a firm’s stock exceeds the cost it takes to replace its depreciated capital, then the firm should be invested in. Yet this determination is self-fulfilling, as the price of a firm’s stock is determined by expectations of its future profitability.

         Say entrepreneur Jeff has a company that he really wants to get off the ground, and his friend Annie is willing to invest. She’s toured his factory, knows Jeff well, and has faith in his drive and work ethic. She has every reason to believe, based on the characteristics of his business, that this will be a lucrative investment. If, however, she’s under the belief that their friends Abed, Troy, and Brita have the same information about his firm and yet will choose not to invest, she too will forgo investing. This is because she is under the impression that no matter how salient the business model may be, it will never get off the ground if she is the only investor, so it would be a waste of her money to do so.

         Of course, there is no inefficiency from an economic standpoint if she is correct in her assumption that their friends were not planning on investing. If she was wrong, however, and still forgoes investment – and many other potential investors do the same in the aggregate – then a perfectly good firm was left uninvested in and an economic failure has occurred due to an irrationality on the part of an individual who is meant to represent the final bastion of true, efficient, rational decision-making.

         When given accurate information about the aptitude of one’s fellow players, the model of standard game theory that we use to govern simple interactions yields good insight into the best possible next move. When our assumptions about the collective differ from the actual collective, however, game theory produces meaning variant results from those which are observed. It is for this reason that, while these inconsistencies are not a reason to do away with traditional game theory entirely (again, models are meant to differ from the real world), conversations regarding the effective scope and scale of game theory models need to be had. The assumption that all players can and do reasonably assume peak rationality on all players’ part, especially, should be critically reevaluated.

         If nothing else, maybe doing so will help me get a little bit of sleep. That is, until I’m jolted awake in a cold sweat by the inevitable next question:

        Wait… How smart does everyone else think I am?