Applying Game Theory Models to Games (Athletic Contests)

Applying Game Theory Models to Games (Athletic Contests)

By Robert E. Wright, Nef Family Chair of Political Economy, Augustana University for the 30^th Annual Teaching Economics Conference, McGraw-Hill Higher Education/Robert Morris University, Moon Township, Pennsylvania, 22-23 February 2019.

“In theory, there is no difference between theory and practice. In practice there is.” Or so baseball legend Yogi Berra once reputedly quipped. He may have said that -- it sounds like Yogi -- but he wasn’t the first to utter those words, an upperclassman at Yale in 1882 was (https://quoteinvestigator.com/2018/04/14/theory/). Similarly, I am not the first to apply game theory to sport (see, e.g., Mottley 1954) but I hope to add to the discussion by suggesting that game theory’s application to coaching territorial team sports like basketball, football, hockey, lacrosse, rugby, soccer, and water polo, can be powerful, but it is not a panacea and even can be counterproductive. Game theory is often more powerfully applied to various non-territorial sports, including baseball (see, e.g., Weinstein-Gould 2009; Turocy 2014) and volley sports like tennis and volleyball (Lin 2014).

I would not call strategic problems in the actual playing of territorial team sports “wicked,” in the technical sense of “wicked social problems” that have no stopping rule, have solutions that are only better or worse rather than right and wrong, and so forth (Peters 2017). But they are certainly “complex” problems a la Nason (2017) and even “chaotic” a la Fergus Connolly (Connolly and White 2017), who incidentally consults for the Robert Morris University Colonials, among other elite level teams. His hyper-interdisciplinary systems-within-systems approach to territorial sports forms the core of the masters course that I teach at Augustana University on the Business of Coaching.

But Connolly’s approach does not make absolutely clear to readers that successful coaches must engage in strategic competition, where strategy refers to the anticipation of the moves of their opponents and not some vague notion of planning, as in the term “game plan.” Game theory, which of course is intrinsically interesting for many students and a key tool in the honing of strategic sensibilities (Dixit 2005), barely registers in Connolly’s otherwise seminal/ovanal/gaminal 2017 opus, Game Changers.

Two player/team/coach games, especially zero-sum ones, seem like a natural way to apply game theory to sport. The realities of on-field competition, however, quickly reveal the shortcomings of simultaneous one-shot games, much as happened when Chicken was applied to the Cuban Missile Crisis (Zagare 2014) and the Prisoner’s Dilemma (PD) was applied to the actual behaviors of criminals (Khadjavi and Lange 2013). In both instances, it quickly became apparent that PD and other simple games exist within larger game structures and cannot in themselves always satiate real world decision makers. For example, HBO’s The Wire showed that the simple PD description we all work through in class (a la Tucker 1983) is, in the reality of Baltimore’s drug scene, embedded in another game, actually referred to on the street as The Game, where “snitches get stitches,” or worse, creating a payoff structure where players keep their mouths shut no matter what (Cherrier 2012).

Simple games also break down due to the speed of competitive play. Time to think through payoff structures does not exist; reactions must be instinctive to be fast enough to matter. The best that can be done is for coaches and players to think through and model various game scenarios and then drill the rational responses, much as ex-NHL player Nicklas Lidstrom has done regarding one-on-one plays in hockey, and so forth (Lennartsson, Lidstrom, and Lindberg 2015).

Many games applicable to sport have mixed strategy equilibria and hence dissolve into Minimax with random strategy solutions (see, e.g., Flanagan 1998), a fact that the offensive coordinator of the Los Angeles Rams seems not to have fully grasped during the recent Super Bowl LIII. Several studies have shown that minimax predictions do not hold up well in the laboratory (Levitt, List, and Reiley 2010) but do on the field, at least where strategy randomization and outcomes can be precisely measured, as in tennis serves and soccer penalty kicks (Palacios-Huerta 2003). Similarly, McGarrity and Linnen (2010) leverage a natural experiment, the injury of a starting quarterback, to show that NFL football teams play the equivalent of a matching pennies game wherein the defense tries to match the offense’s decision to run or pass and the offense tries to not match the defense’s decision. I suggest that football teams actually run three types of plays -- run, pass, and hybrids like draws, options, play action, and screens -- so a rock-paper-scissors type game might be even more realistic (Spaniel 2011).

In any event, working through mixed strategy examples can be helpful for aspiring coaches to see that randomness can be optimal under specific conditions. That does not alleviate their angst concerning their replacement by, if not just computers, then nerds using computers (Davenport 2016; Jones 2018), but it can help them to overcome behavioral biases like the risk aversion that apparently induces baseball pitchers to throw too many fastballs (Kovash and Levitt 2009), football coaches to punt too frequently on fourth down and to run the ball too much, and basketball players not to attempt as many three-point shots as they should (Fichman and O’Brien 2018).

The best applications of game theory to territorial sports often occur off the field but can still be of immense importance to coaches. Aspects of sport design, a huge arena ably surveyed, albeit over 15 years ago now, by Szymanski (2003), are amenable to game theoretic modeling. How to keep up fan interest is a core concern as it forms the basis for all sports funding, except in amateur pay-to-play leagues, in which case maximization of player and/or parent utility is paramount. Most research suggests that fans want the home team to win, but in a close contest, rendering mechanisms for ensuring something like on-field parity of prime interest. That leads in many fruitful directions, like rules for demoting teams from the top tier, as in European soccer leagues, and drafting new players. Forty years ago, for example, Brams and Straffin (1979) showed, with a simple PD game and four fairly realistic assumptions about complete information and incomplete collusion, that North American-style drafts could lead to Pareto inefficient outcomes. As Syzmanski (2010) has shown, however, many early treatments of competitive balance made unrealistic assumptions about the correspondence between individual athletic talent and team wins. In sum, the sum of individual talent can be greater than, equal to, or less than actual team performance.

The need to maintain competitive balance also raises the largely intractable issue of doping, or the use of performance-enhancing substances (Kirstein 2009). Frank Daumann of the Institute for Sports Science in Jena, Germany, recently (2018) offered a game theoretic analysis of the doping strategies of two athletes that can be easily modified to a scenario where two head coaches must decide whether or not to allow their players to use performance enhancing substances not explicitly banned by the league or conference in which they compete. Benefits of doping include a higher probability of winning and hence of job retention (Fizel and D’Itri 1997), advancement, bonuses, and a burnished reputation. Costs include the price of the substances themselves and reductions in athlete health, both presumably small in present value terms, and the risk of a tarnished reputation (Butler 2014). This, Daumann shows, could be modeled as a one-off PD such that both coaches will decide to allow their players to dope although both teams would be better off if they did not use performance enhancing substances.

But of course in team sports, especially the territorial ones considered here, athletes may try to free ride on their teammates. In other words, simply because a coach signals that athletes may dope does not mean that they will choose to do so as players may hope that enough of their fellows will bear the costs of doping to improve team performance without having to bear the costs themselves.

The prospect of free-riding raises the specter of coaches forcing their players to dope, or leveraging asymmetric information to trick them into doping (Johnson 2003), either of which would radically change the payoff structure the coach faces as he or she may have to bear all of the cost of the enhancement substances and any negative social and reputational effects if league officials, competitors, or fans discover the doping, which seems more likely if the coach forces it than if he or she simply allows it (Dunbar 2014). Coercion seems unlikely, moreover, because team sport coaches regularly face free rider problems in a variety of areas and have techniques for mitigating them analogous to the techniques used by those drug dealers in Baltimore that I mentioned earlier.

Like the leaders of drug dealers and criminal gangs (Spergel 1990), organized crime “families” (Shvarts 2002), and most military units (Rose 1945-46; Montgomery 1946), coaches reduce free riding by creating a culture of trust, an ideology of service to others, and pseudo-familial bonds through various rituals and shared adversity. In effect, they try to reduce the economic rationality of their athletes by convincing them that they love their teammates more than they love themselves (Connolly and White 2017), thus inducing them to place a large weight on their colleagues’ well-being in their own utility functions (Bergstrom 1997). Game theory aficionados will recognize the similarities of this with the Battle of the Sexes, the iconic version of which features a husband who wants to go to a football game and a wife who wants to go to the opera but both prefer that outcome only if they can persuade the other spouse to attend with them (Hahn 2003).

Of course, cultural manipulations sometimes fall short or break down under stress, so coaches have other tools for reducing free riding. One I call the wildebeest solution after a technique that wildebeest herds reputedly use to cross crocodile-infested waters during their great migrations across the African savannah. They force the putatively oldest, weakest, least fertile member of the herd into the waters first. As the voracious crocs devour her, the younger, stronger, healthier members of the herd safely cross (Sapolsky 2017). No strategic choice is involved as the wildebeests force one of their number into the river first, but costly signaling may be involved. The key to survival is to appear not to be the oldest, weakest member of the herd or, in our case, team, where death-by-crocodile is substituted with getting cut from the team. Coaches, in other words, can reduce defection, free riding, and shirking by making it clear that players who do not signal that they are “team players,” who do not stand ready to forgo the temptation to free ride, may end up making the ultimate sacrifice (Spence 1973).

Despite some recognition that Braess’s paradox may apply to territorial team sports like basketball (Skinner 2010), superstar athletes know that coaches cannot costlessly bench them, much less cut them from the team, so some may shirk or defect by going for individual statistics instead of wins (Berri and Krautmann 2006; Krautmann and Donley 2009), which is why it is wise to make team performance, short of championship bonuses, a major component of elite athlete compensation (Frick 2003). For most players, though, fear of being labelled the wildebeest is enough to induce them to take one for the team, even if that means injecting or imbibing some new or unusual substance not yet banned.

In sum, coaches and other sports administrators can leverage the insights of game theory to improve competitive outcomes but they need to employ theory carefully and switch toolkits when need be, or face becoming the expendable wildebeest themselves.

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