Let’s attempt to understand the payoffs and coach Spoelstra’s options for game 4 of the NBA eastern conference final (ECF). Before we get into the arguments, here is a brief primer on the subject that we are discussing today.
The 2022 NBA ECF
And the matchup is between the Miami Heat and the Boston Celtics, with the Heat leading 2-1 at the end of game 3. Game 4, just like game 3, is at Celtic’s court, TD Gardens. So, there is a homecourt advantage for the Boston team. Heat’s star player Jimmy Butler just got injured (knee inflammation) in game 3. Let’s assume the injury was not a serious one, and there is a possibility he could be back for the next game. After game 4, there is a maximum of 3 more games, two of them in Miami’s backyard. Whichever team reaches four wins first will win the conference and advance to the NBA finals.
Butler brings advantage, and so is home.
Let’s write down the key assumptions and payoffs. If Butler plays, his team gets a boost of about 0.2 probability points over not playing. i.e., at home, it is 0.6 vs 0.4, and away 0.4 vs 0.2. If he plays in game 4, there is a 0.5 chance of aggravating his injury, making him unavailable for game 5. If he doesn’t play game 4, there is a 0.8 chance he plays for game 5 healthy, thanks to two additional days of rest. If the Heat wins game 4, it will be a huge boost to win the conference, as they have two home and one away matches to realise just one more win. If they lose game 4, it is still fine as they tie at 2-2, with two more to win with two home matches at hand.
What should Spoelstra do?
Well, he should weigh down factors and write payoff matrices and expected values. I will make one, not exactly a payoff matrix, but still capable of describing winning and losing with and without Butler.
These payoff values are arbitrary, but a win without Butler ranks the highest as he will be available as a fitter player for the rest of the games to close out. Butler playing and losing is risky as there is a higher chance of worsening the injury. And the other two results yield somewhere in between these two extremes.
Dominant strategy
If you compare the first column of the matrix with the second, i.e., comparing Butler playing with not playing, you will note that + 200 > + 100 and 0 > – 100. So, under these payoff values, Butler not playing is the dominant strategy.
Expected values
Let Vplay be the expected value for the Heat when Butler plays, and Vno-play when Butler does not play.
Vplay = 100 x 0.4 + (-100) x 0.6 = – 20
Vno-play= 200 x 0.2 + 0 x 0.8 = 40
Again, Butler not playing has the higher expected value.
Any doubts, coach?
The value of the above arguments is only worth the underlying assumptions, which, at this stage, are only arbitrary or speculative.
Tailpiece
(added on May 24, after the end of match 4). Butler played for the Heat, but the Celtics won by 20 points. Butler scored 6 points in the game (his previous scores were 41, 29 and 8). Whether his involvement in the match affected his fitness for future ties remains to be seen or will never be known.
