The Game Theory of Stacking Matchups in Your Favor

Baseball pitchers and competition

San Diego pitcher Jake Peavy did not like being called soft. On August 13, 2008, one day before a scheduled start, Peavy was brooding about accusations that he was choosing to face weaker opponents. Radio hosts were critical that Peavy had missed facing elite pitchers like Johan Santana and CC Sabathia due to an abnormal resting schedule.

Peavy channeled his anger in his next start and outdueled all-star pitcher Ben Sheets for a 3-2 win. Peavy said of the victory: “What a big day for the guys to come back and beat a guy like Ben Sheets, especially after he’s been so dominant,” Peavy said. “Any time you match up with somebody like that, you know runs are going to be at a premium. Those are fun days and days you’re not going to forget.” (source)

In baseball it is very common for top pitchers or “aces” to match up. Great competition is part of the reason because it’s fun to watch closely contested games. Additionally, pairing up similar talent levels ensures good competition across a three or four game series.

Such a situation begs to be analyzed by game theory. Is it possible to game or “stack” the pairings by lying about who’s your best pitcher? What happens if the other side tries to lie too? Does anyone try to do this?

Stacking the matchups and lying about your best

Stacking is based on the principle of saving your best resources for winnable games. If there’s a game you are unlikely to win, for instance, you would rather insert a bad pitcher and save your best talent for another game.

Here’s a textbook example of where stacking could be used. Consider two baseball teams (A and B) playing a three-game series. As is customary, each manager has to assign a different pitcher to start each game. The typical matchups involve pairing pitchers of similar talent. But imagine that team A has a pitcher that is unbeatable and is expected to start on the very first game.

How should team B respond? It would be a waste for team B to use its best pitcher–the outcome will be a loss. Clearly, team B should use its worst pitcher. Consequently, team B will have its first and second best pitchers to face team A’s second and third best pitchers. This stacking should improve team B’s chance of winning the series.

My own experience with stacking

I admit the example is concocted, but such situations do occur. I faced a very similar situation in my sporting experience. I was playing Intramural table tennis at Stanford on a three-person team. We were a good team and we wasted no time in reaching the Quarterfinals. There we faced our first real challenge. The opposing team had a player who was nationally ranked. (His reputation preceded him—we were expecting someone professional like these guys).

It would have been possible to game the matchups. Knowing it would be hard to beat their best, we could have paired our worst player against him and saved our best two players for the other two games.

But in the end of the day, it was our team that got gamed. It was emotions that did us in. The other team’s worst player had been taunting us before and throughout the match. To this day I’m not sure if he did this out of stupidity or calculation, but whatever the case, it worked. We were upset and wanted to beat him convincingly. Our best player rose up to the challenge and took care of business.

Unfortunately that left us at a strategic disadvantage. We now had our second and third-best players against their first and second-best players. We lost against their best player, as expected, and we ultimately lost the other game as well in a close contest. Our tournament run had ended, and our only solace came from losing to the eventual champions.

Stacking is used in baseball

In researching this article, I came across an interesting article from Jeff Sentell of the Birmingham News. Sentell discusses pitching strategy in a high school playoff system. It turns out that some coaches have been stacking the matchups, and it has worked for them:

But [stacking] happens all the time in playoff baseball. It even works. I’d bet many championships have been won across the state of Alabama with choices like the one Vestavia coach Dal Davis used against Mountain Brook earlier this year.

Davis opted to save his junior ace for the second game rather than go head-to-head against Payton Gardner of Mountain Brook. Gardner signed with UAB and was a great option for the first game of any round of the playoffs.

“Gardner is great,” Davis said. “We just wanted to offset him. We threw a quality guy against him and had a chance to beat him and come back with our ace in the second game.”

Vestavia lost Gardner’s game in the ninth inning and won out in the rest of the series. Davis made a play that worked. [source]

It’s risky business applying game theory in sports, but this time the gamble paid off and the coach was rewarded.

What happens if everyone tries to stack?

If stacking works for one team, that raises the question about how the other team could respond. If Vestavia gained an edge by stacking, then Mountain Brook could reclaim that edge by stacking to counter. If decisions can be changed indefinitely, the game might not have a logical end.

But since coaches have to choose at the same time, something more interesting happens—assignments end up being made randomly.

The end result of stacking: pairings end up random

From what I can tell, most coaches don’t try to stack lineups, and even those that do rarely face opponents that respond. So we’d rarely expect to see random assignments. But interestingly, they do happen.

The example comes from high school tennis. Coaches are supposed to seed players by talent level, and the seeds consequently determine matchups. Some coaches have been insincere and lied about seeding to stack matchups. Apparently other coaches have responded in kind to create a fun game of mind reading.

What has eventually happened? Here’s the scoop from a high school tennis system in Texas:

The Galveston County Daily News recently reported that several class 4A and 5A high school tennis teams in Texas have removed player numbering in their leagues. Dual team matches “will have the same win 10 individual matches, win the overall dual match concept, but with an added twist-no true lineup.” This new format creates the type of payoff matrix where [the random equilibrium] occurs. Assuming 10 slots for individual player matches, each can choose any combination of players to fill slots 1-10 but they do not know if the other coach is playing Sincere or using some other Insincere combination of players. Thus, there is no incentive with this game format except to randomly select players to fill up the 10 spots. [source]

Stacking can be hard in practice

In practice, stacking gets complicated due to additional considerations. For instance, I was thinking about applying stacking to the baseball playoffs, and I consulted two of my most knowledgeable friends. They were concerned about human factors like accounting for possible pitcher reuse on short rest. Just adding this assumption raised the complexity immensely to lead my friend to consider using Monte Carlo simulation. And this is without taking into account other things like the chance of injuries or historical data on pitchers. As you can see, it gets complicated very quickly!

But I hope the examples give you ideas about stacking matchups in your own life. Imagine your company is competing with a rival for a series of projects. How should you allocate talent during your company marketing? This is a game of seeding. Your opponent will likely put their best talent for the most profitable project. Do you compete against them, or try to run the table for the remaining projects? How do you account for repeated play? These are the issues the best managers and partners know how to handle.

Key lessons from stacking:

• It’s possible to improve winning odds by stacking
• One strategy is to save the best talent for winnable games
• If your opponent starts countering, you will likely end up randomizing
• Practical constraints, like emotions and extra rules (ex. some pitchers can be reused), need to be addressed