Mind Your Decisions

Truels, or How Game Theory Predicts Survival of the Weakest

Every Tuesday is a Game Theory article at Mind Your Decisions.

In my recent game theory post, I wrote about how seemingly worse employees can get on better projects–if you didn’t read it, please do as the answer may surprise you. This strategy really works, and as Joon and Rohomech pointed out in the comments, they implemented the strategy by incorporating it into their reputations.

This week I will continue discussing reputation and talent and how they can affect your odds of succeeding in a winner-take-all battle. I specifically will analyze the truel, which is a three-person analog of a duel.

(Incidentally, the movie The Good, The Bad, and The Ugly has a fantastic truel and is a great example of how game theory works [warning: movie spoilers].)

For my analysis, consider the following truel that I’ve slightly modified from The Ultimate Puzzle Site

Adam, Bob and Charlie engage in a truel with the following rules:

• Shooting order is determined by drawing lots randomly.
• They continue firing at each other in this order until only a single person is alive.
• All shots are lethal, and everyone knows that Adam hits in 100% of all shots, Bob hits in 80% of all shots and Charlie hits in 50% of all shots.
• Each person chooses his ideal strategy.
• Each person has the option of intentionally missing.

The Question: Would you rather be Adam, Bob, or Charlie? In other words, who has the largest chance of surviving the truel, and how big is this chance?

To start analyzing, it is worthwhile to think about a duel. In a duel, it is pretty obvious that both higher marksmanship and shooting first help your winning chances.

Does our same logic work for a truel?

Amazingly no! In fact, both conclusions are false. It turns out that the worst shooter, Charlie, has the best chances of surviving. Almost as surprising, this is true regardless of the shooting order.

The reason is that the good shooters Adam and Bob know they are more of a threat to each other than Charlie is to them. This means they both prefer to fire at each other until one of them dies. Therefore, Charlie will always survive until only two people are left AND he will be the first to shoot. This happens regardless of the shooting order.

One noteworthy case is when Charlie is the first shooter. In that case, he does not want to risk killing Adam or Bob since he would then have the second shot in a head-to-head face off. His best strategy is to miss intentionally and let Adam and Bob fire at each other. Accordingly, Charlie will end up in the two person face off with the first shot.

Another stunning result is that the winning percentages between Charlie, Adam, and Bob are not even close. The worst shooter, Charlie wins in an amazing 52% of the truels. The best shooter Adam wins in only 30%, and Bob has the worst winning chances at 18%.

This is an odd survival of the weakest.

The full proof is listed in the puzzle solution though it takes some concentration to follow.

Now, this game sounds strange, but I suggest it is not merely a mathematical construct. I can think of a couple of situations where truel-like behavior occurs.

The Weakest Link:

The game starts with several players answering trivia questions to generate the total prize money. After each round, contestants vote to remove a player.

In early stages, knowledgeable players are not voted off since they answer lots of questions and raise the prize money. But as the game gets to fewer and fewer contestants, and starts to resemble a truel, the stronger players start voting each other off, and it is often a weaker player that holds the deciding vote and ends up in the final head-to-head match.

Jeopardy:

Suppose you have $2,000 and your opponents both have $20,000 and there is $6,000 left on the board. What is one possible strategy? Do nothing! I think one of your best chances for winning is to keep your total positive (avoid wrong answers) and hope that first and second place engage in a bidding war during final Jeopardy with wrong answers.

You don’t have many choices, so just stay out of it and make them forget you are even there. It is rare that the third place total ends up winning, but the times I’ve seen it happen are because first and second place bid too high against each other.

Key Lessons from Truels:

1. If you find you are the weakest in a winner-take-all competition, don’t do any thing until you are head-to-head in a duel.

The bigger competition will likely fight each other and consequently improve your odds when you do act.

2. You can do even better if you actually are very talented but get a bad reputation.

If Charlie were actually a great shot, that would amplify his winning percentage. The bad reputation keeps Charlie safe in the truel, and the good marksmanship improves his chances in the duel.

I imagine this is how people win those winner-take-all reality shows: they are non-threatening in early rounds to eliminate competition and then crush their opponents in the final round.