Studying games without game theory

Journal of Sport & Exercise Psychology has an experimental paper on soccer penalty-taking. The design is supposed to measure differences in perception and performance in penalty-takers under under stress and no-stress conditions. A bunch of players were asked to take a bunch of penalty kicks each; the control group were told that nothing depends on how often they score, whereas the treatment group were rewarded for performance in various ways, monetary or otherwise. Three things were then measured for penalty-takers in both groups: what they were looking at before taking the shot, how long they were looking at it, and how well they did (i.e. frequency with which they scored). The experiment finds that members of the treatment group were more likely to concentrate their eyesight on the keeper before the shooting more, and performed worse than members of the control group.

This is a flawed research design.

The flaw is to take into account only the psychological aspect of the penalty shooting game while completely ignoring the game-theoretic aspect. A penalty kick is a simultaneous-move game between the taker and the goalkeeper: the taker decides where to place the ball, and the keeper decides which way to dive, both at the same time. Moreover, it is a game in which neither the taker nor the keeper can afford to stick to one choice all the time: if the taker only shoots near the left post, for example, the keeper will figure it out and start diving left all the time; similarly, if the keeper dives only to the right, the taker will start shooting left. (In game theory lingo, the game has no pure strategy equilibrium.) However, it does have an equilibrium in randomized strategies (mixed strategies). What this means is that each player picks a certain frequency with which he plays each strategy; for example, assuming for simplicity that you can only shoot left or right, the taker chooses to shoot left with some probability p and right with some probability 1-p, whereas the keeper chooses to dive left with some probability q and right with some probability 1-q, where p and q are both strictly between 0 and 1. In other words, both players realize there is no single strategy that works all the time, but that there can be a certain mix of strategies that can work some percentage of the time, just like poker players realize that there is no strategy to play any given hand that's always correct, just a strategy that's correct on average. What's more, once you choose to play equilibrium frequencies, you score with a certain fixed rate no matter where the keeper dives (if you're the taker), or save with a certain rate no matter where the taker shoots (if you're the keeper).

The sports psychology paper is missing all this. As a result, we have no idea whether players that perform better do so because they have been placed under "no-stress conditions" or because they are closer to equilibrium frequencies than their counterparts. The correct design would be as follows. First, you calculate equilibrium shooting frequencies (here's one attempt at this). Then you proceed with the treatment/control design and let your players shoot enough times to be able to tell if they're converging on the equilibrium frequency or not. Those who do not will unfortunately have to have their datapoints thrown out: you won't be able to tell if they're doing well/badly due to control/treatment or due to deviating from the equilibrium. However, if you see the difference in performance between treatment and control groups among players that do follow the equilibrium strategy, you can be sure that that difference is really due to treatment.

I have a suspicion that if the equilibrium variable were accounted for, the significance of results from this paper would disappear.