A Most Interesting Truel

A Most Interesting “Truel”

So, already the title is a misnomer. What I am going to write about is probably not really at all a truel. A truel a three person duel. From this article in Mathematics Magazine, a truel is characterised by situations where there are 3 people particpating and the variables include:

“*the probability of each player hitting their chosen targets (often not assumed to be the same for each player)

*whether the players shoot simultaneously or sequentially, and, if sequentially, whether the shooting order is predetermined, or determined at random from among the survivors;

*The number of bullets each player has (in particular, whether this is finite or infinite);

*whether or not intentionally missing is allowed.” Read more in wikipedia.

Anyways, it occurred to me that I took part in a very interesting social experiment several times a week. From a game theoretic point of view, it is really quite fun to think about and try to model. I’m of course talking about cutthroat racquetball. If you’re unfamiliar with racquetball, it is sort of like tennis- you can learn the basics here. One difference is tennis can only be played 1v1 or 2v2. Racquetball can be played 1v1, 1v2, or 2v2.

Here are the basic rules for 1v2, or “cutthroat”.

(1) Set a service rotation for all three players. This rotation does not alter.

(2) Start the game with the server facing off against the other two players.

(3) Play as if it’s doubles when you’re serving.

(4) Use a singles strategy when you’re part of the duo returning serve.

(5) Alternate hitting the ball between the solo player and the doubles team during a rally.

(6) Score points only when you’re serving.

So, more appropriately, cutthroat is really a long series of three different 1v2 games. Maybe an example will help.


Sam (S), Larry (L) and Donny (D) are playing cutthroat.

Using some method, someone is selected to go first. Today, S will serve first. S goes to the front and enters the server’s box. L and D go to the back wall and prepare to return the serve. L and D work as a team and play until S can’t return the ball or misses the shot. Then L move to the serving position and it is S and D in a team against L. The same procedure repeats as D moves into the serving position and L and S work as a team.


So, what is interesting here? It is the fact that it is a complicated mixed motive  game where you teammate changes quickly and your ‘pure conflict’ and ‘pure cooperate’ strategies also change many times within a given game.

In the macro view, the game is simple like any other game: you want to win. You cannot share the ultimate victory, only one player wins in cutthroat. With dotted red line <——–>representing pure conflict interactions, the game looks like:


The reason why this is worth writing about is the three “subgames” that occur in any one game of cutthroat. With solid blue line <——-> representing “pure cooperation, we get:

So literally within the span of one or two minutes, you can be teammate and then mortal enemies with the same people! This means you can’t get too excited and brag a lot when you get a “kill shot” (there are a lot of similar duel and racquetball terms…) that ends the server’s turn, because the server will be your partner soon enough. If you’ve made them angry, they may be less likely to want to cooperate with you even though you are both supposed to be playing pure cooperation strategies.

Also, there can be sabotage and intrigue like any game. If you are playing to 15, and the scores are L=13, S=10, and D=4, D is really at a strategic position. D probably cannot win at this point. But, D can be decisive as to who will ultimately win. D could conspire with S and “miss” a few returns and allow S to win when L otherwise would have.

But, as cutthroat racquetball is a often an iterated truel you play week after week, this conspiracy could prove devastating, as L will probably noticed the intrigue and D will have no friends after a short time.

But, nothing stops L from making the same arrangement for sabotage with D. Moreover, if D is frequently in third place with L and S competing for first, and D has a reputation for throwing games, L and S should have serious doubts about whether or not D will continue to follow through on their conspiracies. If on Thursday D throws the game for L, L could reason that D doesn’t really care about the game and could very well throw it for S on Saturday.

(It is sort of the reasoning on why you should be careful in trusting a double agent. If they are fine with being a double agent, how can they make a credible promise that they are not a triple agent (someone actually working for the enemy, but acting like they are working for you by “working” for the enemy to get secrets). As reference, see the infamous Nina Myers from the TV show 24 as well as this list of fictional double agents.)

At any rate, D will probably never get better if s/he constantly engages in intrigue, and probably find a statistically significant increase in the number of times when s/he is “accidently” hit by the ball from behind. Ouch!

But the game theory doesn’t end there!


We can factor in a couple of truel concepts. S, D and L might not all have the same skill level. And there could be varying degrees of knowledge about those skills levels (D could know how good L is, but not S for example). Although, skill level tends to become readily apparent after a few matches. And they are iterated as (hopefully) no one dies from racquetball as opposed to a gun slinging truel.


There is a surprisingly huge (996!!) number of articles that return when “tennis” “serve” “game theory” is searched for in google scholar when restricted to social science and econ journals. Maybe 1/10 of those actually reference tennis serves with respect to game theory, but still, ~99 is a lot.

A disappointingly small (0) literature relative to racquetball and truels, or racquetball and serves exists. But what applies for tennis serves is generally applicable to racquetball. See for example:

Minimax play at Wimbledon

On the advantage of serving first in a tennis set: four years at Wimbledon

The Tennis Coach Problem: A Game-Theoretic and Experimental Study

Some remarks on strategy in playing tennis

There is a lot to be said on how you should serve from a game theoretic  perspective. It might not always be to your advantage to serve your best serve every time. Sometimes your serves and shot selection should be random. Randomness and degrees of randomness can all be nicely factored and written out.


This can hardly be overstated. Both in the sense of “hand-eye” and the two defensive players. It is a perfectly legitimate play to serve a ball straight down the middle with the expectation that either (1) neither defensive player attempts to return because they think the other is going to, or (2) that both go for the return, crash rackets, and lose the point. And, racquetball is fun because of its speed and requirement for quick reactions, so there is not a lot of time to communicate and strategize in game. Theoretically, the game is sequential- going from offensive server’s play to the defensive team’s play. But, fast play can lead decisions to be made near simultaneously.

What can we make of any of this? I think the most interesting thing is to note how even in a overall pure conflict game there are sub games that require cooperation. It is virtually impossible to win cutthroat without cooperating when you are on defense. And cooperation equilibriums always occur. Even in a game where you are still trying to win! Even when the “prize” cannot be shared! Even when someone can make you really mad by ending a “hot streak” (which apparently don’t happen) of points!  Maybe the racquetball truel is a good model for fostering (focing?) cooperation from opposing players in a multitude of settings. I am interested in cooperation, conflict, and coordination; all of these I find about 5 times a week in a most unlikely of places: the racquetball courts.


Posted on June 17, 2011, in Uncategorized and tagged , , , , , , . Bookmark the permalink. Leave a comment.

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