Game Theory in Basketball
In order to score points in basketball, there are three main ways: free throws (worth 1 point), two-pointers (worth 2 points), and a three-pointers (worth 3 points). For an average player in the National Basketball Association (NBA), their free throw scoring average tends to be the highest (77.8%), with two-pointers second (46.6%) and three-pointers third (36.7%). These scoring percentages vary as free throws are unguarded by defenders, two pointers are guarded but closer to the net, and three pointers are guarded and are shot further from the net. During a game, coaches draw up plans to score as many points as possible given the varying difficulties of the shots and different defenses from the opposition. The “plays” become especially important when the score of a game is close during the final minutes of the last quarter. A common scenario in basketball occurs when a team has possession of the ball and are losing by 2 points with not a lot of time left in the game. Coaches will usually call a timeout to create a “play” for his/her players to execute.
Since free throws depend on the other team’s players fouling, they are out of the equation so coaches only consider two-pointers or three-pointers. While two-pointers are easier to score, the game would be tied if scored and the game would go to overtime. This can be a problem as basketball is defined by momentum and “hot scoring streaks” and depending on which team has the momentum in the final minutes of the game, that team can carry that momentum and dominate the overtime period. Three-pointers are harder to score but the game would be won if scored. The opposing team also has to make an important decision as defending a two-pointer vs a three-pointer requires substantially different formations that would leave the team vulnerable to the other shot.
According to Chow et al., concepts relating to game theory can be applied to this scenario as it is basically a simultaneous game where attackers have to decide whether to go for two or three points while defenders have to decide whether to defend the two-pointer or three-pointer. This is similar to the penalty kick example covered in class since the two “sides” have to pick where to “attack” and where to “defend” and even if the attacking team makes the right decision, there is a chance to miss while even if the defending team makes the right decision, the attacking team can still score. As mentioned in class, a dominant strategy is a strategy that is the best response to any other strategy. In this case, there is no dominant strategy as there is no single best response to all the opponent’s strategies. For the attacking team, if the defending team defends the three-pointer, the best response is shooting a two-pointer while if the defending team defends the two-pointer, the best response is shooting a three-pointer. Likewise, if the attacking team shoots a two-pointer, the best response for the defending team is to defend the two-pointer while if the attacking team shoots a three-pointer, the best response is to defend the three-pointer. Since there is no pure Nash equilibrium, there must be a mixed strategy Nash equilibrium which depends on how well each player shoots and how well defenders defend. The writers of the paper used average shooting numbers and assumed how these shooting percentages change depending on which type of defense they are facing. After running the calculations for mixed strategy equilibriums, they determined that the “shooter should always shoot the three if the defender defends against the three pointer less than 62.6% of the time” while the “defender should always defend the 3 if the shooter shoots the three more than 34.6% of the time.”
Another scenario the article studies is when a team is down by three points instead of two. This scenario is a little bit more complicated since the attacking team can quickly score a two-pointer, in which point they will be down by 1 point. Then, they can foul the opposing team, hope they miss one of their free throws with which the team would get the ball back and score a two pointer to tie or three pointer to win. If the opposing team misses both free throws, the team gets the ball back and only needs a two or three to win the game and if the opposing team scores both free throws, they need to score a three to tie the game up and send the game to overtime. The defending team will have the same strategy of either defending the two-pointer or three-pointer. Chow et al claims that despite the much more complicated options for the attacking team, this scenario is also a simultaneous game that can be analyzed with concepts from game theory. Again, there is no pure nash equilibrium for similar reasons as the above scenario where one team can just respond to the strategy of one team by going for the strategy that exploits that weakness. Chow et al claims that since shooting the quick two and getting the ball back after the opposing team’s free throws is the only way to win the game during the regular period, the attacking team will only shoot a three 29% of the time. On the other hand, since the two point option is a lot harder to pull of successfully, the defending team would rather defend the easier three-pointer strategy 74% of the time. Thus, there is a mixed equilibrium for this scenario as well despite the much more complicated options for the attacking team.
http://faculty.haas.berkeley.edu/rjmorgan/mba211/Chow%20Heavy%20Industries%20Final%20Project.pdf