# The Buzzer Game [closed]

Two players, A and B, are each given air horns. Also present is a device which will at some random point produce a noise. If the device produces a noise before either player, they die. If both players produce a noise before the device, the first to produce the noise is killed and the second one lives. If only one player produces a noise and the device sounds before the other player gets a chance, the player unable to produce a sound is killed. Is it necessarily so that if they are perfectly logical and selfish beings, they both shall die?

• I remember seeing this situation in the Book of Questions, except it was you and a significant other, and it was a moral decision rather than a game-theoretical one. – Joe Z. Sep 21 '15 at 15:31

Since there is a random element, logic would seem to fall out of the equation. They are both going to try to survive, which means blowing their horn as soon as the other person blows theirs or blowing it right before the device sounds. Since human reaction time is quite fast, the odds of guessing when the device will sound accurately enough that the other person doesn't have time to respond are very low. So, then, the best bet is to wait for the other person to blow their horn, right? The hitch is, they are both perfectly logical so they know that the other one has come to the same conclusion. Then they will both know that waiting for the other to blow is certain death for all, so they will either blow their horns randomly or try to trick the other person into blowing their horn. In a situation like this, tension would be very high, and making a sudden movement might be enough to get the other person to blow first out of overactive reflexes. Assuming that they are not exactly the same person, despite being perfectly logical and selfish, one may succeed in tricking the other into blowing, or one may blow first in hopes that they are blowing just before the device goes off.

They don't necessarily both have to die.

By the given parameters, A and B are both logical and selfish. Logically, there is no single-sided solution guaranteeing a logician's own survival - sounding the horn immediately has the same deadly effect as waiting indefinitely for the other logician's horn. The 'selfish' parameter is therefore disregarded as (logically) a 'don't care' condition. The remaining thing to consider is the survival of at least one of the logicians (in this case, the other logician), which each can effect by sounding their own horn first.

Both logicians follow the above logic and immediately sound their own horns. The possibility of survival arises when the killing device does not sound in less time than the logicians are physically capable of sounding their own horns.

Assuming identical reaction times, speed of thought, etc, the immediate simultaneous sounding by A and B means that none of the deadly conditions comes into play, so both logicians survive. If that assumption doesn't hold, then the slower logician survives.

• I see what you mean by saying that selfishness is thrown out by the inability of logic to save oneself, but I don't think that it would lead to immediate altruism. As long as there was still a chance at survival, I believe the selfish logicians would still pursue it. Although, sounding one's horn immediately might still be the best course of action to maximize your odds of not getting killed by the device. – Warkgnall Sep 20 '15 at 8:32
• @Warkgnall It isn't that there is a slim but elusive chance of survival: without being able to control the other party, there is no strategy that leads to any chance of (their own) survival. It is arguable that being able to predict the other party's actions can be another lever, but that leads to the same solution: immediately sound the horn. Even there, 'immediate' is impossible to rely on due to those tiny random variations between A and B. So, strangely enough, the only chance of survival arises from a motivation of self-sacrifice in this set up. – Lawrence Sep 20 '15 at 8:42
• I like this answer best, however it does assume a reflexion time set before the start of the 'game' for each logician to arrive to this conclusion. If they do have that time, it seems to be as it being a given that both will immediatly sound their horn instantly at the start of the game, as it is the scenario with the most survival rate by a clear margin (being one or both of them) – Spacemonkey Sep 21 '15 at 20:08
• Thanks @Spacemonkey. You can avoid the reflection time by either assuming A & B are identical (same time to come to the same conclusion), or just default to the slower one having a chance to live. In either case, the possibility of living arises when the device doesn't sound before A & B. – Lawrence Sep 22 '15 at 10:05

Solving such questions is easy using the method given here.

The three strategies given to the players to choose from are as follows:

1: Never horn
2: Horn immediately
3: Horn after the other player

The payoff-table looks as follows (1 means life, 0 means death):

B adopts 1      0,0               1,0             0,0
B adopts 2      0,1             1/2,1/2           1,0
B adopts 3      0,0               0,1             0,0

The only Nash Equilibrium is when one player chooses 2 and the other chooses 3. Meaning that immediately horning, or horning after the other player are both equally good strategies. Hence they will randomly select strategy 2 or 3.

It depends on what happens if they're indifferent. Assuming symmetry is broken somehow (because if it isn't, then they trivially both die), then one person will sound their airhorn first. Then, the second person will die either way, but they get to choose if the first one lives. The question doesn't say anything about what will happen then.

• sorry about the confusion, i fixed it – el riddler 73 Sep 20 '15 at 2:46
• @elriddler73: This is the answer to the updated version. – Deusovi Sep 20 '15 at 2:47

Firstly, the question did not resolve the situation when one player sounds the horn during the time the device is sounding. Does that player die from being the first person to sound the airhorn, or not die (while the other dies) because he "produced a noise and the device sounds before the other player gets a chance"? If he doesn't die, we have a loophole.

Secondly, the question does not specify if they are in the same room, and if they are, whether the device itself is present in the same room. If they are in separate rooms and cannot see the device directly, we note the question does not specify if the players have heard what the device sounds like before (and when it does, where the sound comes from). Then similar to Warknall's answer, they can try tricking each other that by making it sound like the device has sounded.

If they are in the same room, the question does not specify what qualifies as a "player making a noise" means. I think the intended answer to this is making a sound from their respective assigned airhorns (otherwise they would have a hilariously silent tickle war). If so, then the logical thing to do is to grab the other player's horn and sound it as soon as possible. Alternatively, they can cooperate and try to destroy the device together, but I doubt that is an option within the framework of the question.

It is not necessary that both die. As both are perfectly logical both know that waiting for the other to make a noise will kill them both. Therefore both can agree to adapt the rules to make one of them survive, assuming equal chances for both.

One possibility is to prevent that both players can hear each others noise. For example by putting on headphones with loud music. Of course they both will make sure the other can't cheat, by checking the headphones before.

• Putting a headphone on might actually kill them both. They will, after knowing the other cannot judge by the ears, tries trick each other by acting it, resulting in fails to blow before the device buzz. – Alex Sep 22 '15 at 16:00
• @Alex The question asks if it is sure that both die. I tried to show that this is not the case. – Sleafar Sep 22 '15 at 16:06