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.
Update: Here is another write-up using computer code for Markov chains: Truel simulation and results.
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.
I’m curious how the truel applies to a game like Texas Hold’um Poker – it’s not usual to see the third place guy in chips win at the final table.
Well, the problem with poker is how the chips get distributed when the two top players finish their duel. Unlike the gun-fight, the winning poker player *increases* their resources. And with a larger chip count, their chances of winning go up even more.
However, in the shootout, the ammo in the guns is limited, so after the best shooters finish their fight, the winner might be out of ammo, or worse the winner could even be injured.
@Poker discussion: Poker is different because the winner gets more chips.
Come to think of it, I think a more appropriate analogy is the board game Risk. A third player, who perhaps owns Australia, is able to surmount a comeback if the two bigger players squabble over Europe and Africa. The third player just collects armies, stacks up cards until the trio is worth a lot, and then makes a grab for Asia.
@Presh
I think Joe P was asking for a solution to his poker puzzle as well….though it seems like the weaker player will get defeated.
I know this is way late for the discussion, but another important point in Poker is how an individual round can work out. When a weak player meets a strong one in a head to head duel, it must be noted that he must usually win numerous hands in a row in order to defeat the strong opponent. There is no 50% KO chance like in the truel. The most the leader can lose is the total of the weak player’s chips at a time. This usually makes a comeback very difficult.
Cody: Chip stacks do seem to matter much more in poker, especially since weaker players are more damaged by increasing blind antes = less time to ride out luck.
If Cody thinks he’s late, I got him beat by 1/2 year.
Anyway, I’ve been working on truels ever since I heard the word 11 days ago. I eventually found your site and noticed the percentages for the two weaker shooters differed from those given in the problem I was given. This motivated me to generalize the computer program I wrote which simulated the situation (it now accepts the shooters’ percentages from the keyboard). I only addressed the case that the first and weakest shooter intentionally misses on his first shot. The mathematics which analytically solves this situation and a computer program which simulates it may be seen if you follow the thread at
http://tobee-interpres.blogspot.com/2009/08/zee-da-gud-tobee-da-badd-and-luna-da.html
The simulated probability that the weakest shooter will survive matches up well with the analytically determined probability.
It was fun thinking about this problem and, like most computer programs, took many revisions to get it “just right.”
I can’t believe I read this last year but only now did I think of this.
In regards to the shooting truel with Charlie being the first shot, I think it not only behoove him to purposely miss, which you said, but to never shoot at Adam. The reason is that there’s no guarantee that Adam and Bob are completely rational, and being shot at would probably put Adam on the defensive and want to kill Charlie even if he’s not a big threat. Same would go with Bob, except Bob is not a guaranteed hit, so shooting and missing Bob first might work.
I wonder if Charlie should just obviously shoot so badly that both opponents have no suspicion as to his bad shooting (shooting very wide) or if he should show them both he refuses to shoot first by shooting into the air or putting his gun away. No point in wasting ammo after all, especially if it’s limited. Though this might draw suspicion.
Also, I’m terrible at poker, but often manage to knock out a few players because I’m not a big threat, and have managed to win a game…sometimes people let me get chips because they know they’ll get it back from me.
On the truels solution, since they have the option of missing intentionally, isn’t the optimal strategy of surviving for all 3 participants to intentionally miss each shot? That way the game continues forever with a 100% chance of survival for everyone.
Rsg, you are assuming that the payoff for staying in the game forever is higher than trying to finish it staying alive. A reasonable assumption since the payoffs are based solely on surviving and not “winning”.
Buuut, the whole point in game theory is that if you miss intentionally, you have no guarantee the other players will do the same and honor the “miss intentionally agreement”.
A history of past moves from other players help in analysing their reputation and knowing if they can be trusted, but the first player to shoot has no history. Assuming the worst, the optimal move is to shoot. Then the optimal move for the second player, seeing the first player aggressive history, is to also shoot.
The good news is that in real life, “civilized people” have the implicit reputation of not being aggressive, the “assuming the worst” doesn´t happen, and peaceful agreements can often be made.