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In a game of cheat/bullshit with at least three players, what is the incentive for a player to challenge a claim when he/she does not stand to gain anything in winning the challenge, but risks taking the entire stack of cards if he loses the challenge?

The only situation where I think it would make sense to challenge is the following: If someone just placed his/her last card down and I am the only one who has sufficient information to conclude that he/she is cheating, then it makes sense for me to challenge in order to prevent him/her from winning. But apart from this situation, does it make sense to challenge at all? (Assuming we play be the rules on Wikipedia, where play continues normally with the next player regardless of the outcome of the challenge)

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You can call "cheat" even when you aren't 100% positive someone is lying to keep the fear of being called out real. Like @PhilipKendall said, if you are only going to call "cheat" when you can prove it or when you are at the last opportunity to keep someone from winning, there is never any fear of being called as long as you make relatively conservative plays. And that doesn't make for a very interesting game.

Part of playing cheat is mind games and taking some risks. You have to evaluate your own risk-reward along with the risk-reward of other players.

For example, say I hold two 3s and my two opponents each hold one 3. I could very easily put down the two 3s and be perfectly safe. I could also call three 3s and hope to get away with it. Neither of my opponents can prove that I'm lying based on cards alone (note that I don't know this when I make the play, I need to evaluate the risk-reward of getting rid of an extra card vs. having to pick up the pile). Is it worth it for them to call cheat? Maybe. Is it possible I actually have three 3s? Yes. Is it likely? Maybe, depending on how the game has gone so far. My opponents need to evaluate the risk-reward just like I did. If there are only a few cards in the pile, the risk is fairly low and the reward is that they make me pick up cards. The other reward is that it can make me more hesitant to lie. Which slows me down and makes it easier to win.

What if there are more cards to pick up? Is it worth it to call cheat? Maybe. The likelihood of lying goes up as people start running out of cards and the pile of played cards gets bigger. But the penalty for being wrong is much higher.

The risk-reward analysis gets even more interesting when you consider how close your opponent is to winning. Which is worth more, getting your opponent with 5 cards to pick up 20 or getting the one with 15 to pick up 20? Is it worth picking up 20 yourself if you are wrong? How likely is it that the person with 5 cards is lying about their last play? What about the guy with 15? Same arguments apply to how close you are to winning and if it is worth it.

Another potential reason to call cheat is to get cards. I know this sounds crazy but bear with me for a second. If you aren't doing well and you opponent is, having cards can make it really easy to call cheat on someone and give them cards. And you can take someone from being on the cusp of victory to having as many cards as anyone else fairly quickly. (From a risk-reward perspective, the risk is fairly low to me, since I'm probably going to lose anyway if I don't do something, but the pay out is huge if I can keep you from winning and give myself a second chance, either by calling you on a lie or making it impossible for you to lie in the future and play your last card.)

Personally, I also play a little bit of the agent of chaos role sometimes. I will sometimes call cheat even if I'm not sure. And it can keep my opponents honest, which is to their disadvantage. It makes them think that they could get called any time. So they are less likely to make risky plays and be very conservative. And that helps me.

  • I'll mark this answer as accepted as it answers most clearly in paragraphs 5 & 6 why I should be the one calling "cheat" instead of a "third player" who calls "cheat". I feel that this answer still isn't very clear about this point, but it seems to suggest that I may call "cheat" (instead of the third player) because I am in a different situation as compared to him/her (e.g. by number of cards left or the necessity to cheat in a later turn). But paragraphs 3 & 4 seem to be cases where I should just let a third player call "cheat" on my behalf. – Bernard Nov 24 '15 at 13:50
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Because if you're only ever going to call "cheat" when I play my last card, I'll cheat like crazy on every turn other than my last card and do everything in my power to make sure that I'm not cheating on my last card.

  • Assuming there are more than two people playing, then there are other player(s) who can call "cheat". Thus, you'll still be kept in check by the other players even if I never challenge. I'll thus be able to keep myself safe from challenging and losing the challenge. – Bernard Nov 23 '15 at 3:04
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    Unless all the other players are using this strategy... – Philip Kendall Nov 23 '15 at 7:01
  • Yeah, but this is usually not the case... I was hoping for a stronger incentive to call "cheat" in usual games of cheat from a game-theory perspective but seems like the incentive for calling "cheat" is quite limited. – Bernard Nov 23 '15 at 13:31
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You're correct in that it's usually better to let other people call cheaters out, since there's a personal risk but no personal reward. However, imagine the case where it's about to be my turn, there's a large stack on the table, and I have no option but to cheat. Here, it might be best for me to call "cheat" on the person who just played in order to clear the stack lest I be called out as a cheat after I play. If I'm right, now there's not much of a stack to pick up if I get called out, and if I'm wrong, well, I might have been picking up those cards anyway.

  • True, this is another case where there is an incentive to call "cheat", but it seems to be quite a specific case where you're pretty sure that cheating on your turn will be detected by your opponents. – Bernard Nov 23 '15 at 13:34
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From a game theory perspective, you'd have to analyze this in more concrete scenarios. Yes, if someone says "one ace" on the first turn where everyone else has the same card holding and nobody happens to have three or four aces, it's a pretty silly risk to call them on it - which is why nobody does call them on it in that scenario. (Although, if you have an ace already in your hand, it's not necessarily a risk at all - see later.)

However, there are several factors that influence when you should call a player's apparent bluff.

First off, you need to calculate the apparent cost to you of calling that bluff.

  • How many cards will you acquire
  • Are those "useful" cards, "irrelevant" cards, or "harmful" cards? Useful cards are cards that fill a spot you know you'll need. Irrelevant cards are cards that overlap cards already in your hand (which you don't plan to bluff with). Harmful cards are cards that you wouldn't normally need in the sequence of things, so having them means you're less likely to win.

Second, you need to calculate the cost to you of not calling your opponent's bluff.

  • How close to winning does the player appear to be (particularly, in relation to yourself)
  • Is this a strong player, or a weak player?

Third, you need to calculate the odds of the player bluffing.

  • Does the player have a lot of cards?
  • Have they incorrectly called others' bluffs such that they would likely have cards of this number?
  • How frequently does this player bluff?
  • How many of this card type do you have?

Fourth, you need to calculate the odds of other players calling his bluff.

  • Are there other players who are likely to have a lot of this card type? What other players are likely to have as much information as you about this card type?
  • How close to winning are other players?
  • Do you have a player who likes to call bluffs?
  • Is this a stronger player who others are likely to call simply to prevent him/her from winning?

All of these factor into whether you should call them (and of course, in a perfect world they'd have the same considerations).

There will be scenarios where the odds of them bluffing are high in your estimation, but aren't all that high in others'; and there will be scenarios where your personal incentive (cost-benefit analysis) will be different than others' such that you would choose to call the bluff while others wouldn't, even for the same information level. If I'm in first and the second place player just put 3 of his 5 cards down, I have a huge incentive to call that bluff, compared to the other players (assuming I'm not going to win next round).

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This is actually a variation on the Game Theory problem The Volunteer's Dilemma. Essentially, there is a cost to you calling cheat (The chance of being wrong and taking cards), and a benefit to all for you doing it (Stopping an opponent). There are a lot of different possible equilibria, but if every player acts the same, then the only equilibrium (state where nobody can do better by changing strategies) is when each player calls cheat some percentage of the time, where the percentage is based on the number of players and the costs.

Two take-aways...

  • Why this helps us. Mathematically, as the number of people increases, the optimal amount of time they volunteer decreases. In fact, when the number of people approaches infinity, the chance of any person at all calling goes to 0. So basically, the more people you are playing with, the less often you should call, but there is still such a thing as not calling often enough.

  • Why it's different. In Cheat, unlike a vanilla Volunteer's Dilemma, everybody has different costs. These are roughly based on how many cards you have, and how likely you think the other player is cheating. Even so, there is a mixed strategy equilibrium that involves you calling cheat sometimes.

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This by no means the entire answer (the other parts have been mentioned by other people) but a very good reason to call "Cheat" is...

It's fun!

I do play games to win, but I mainly play to have fun. Calling "Cheat" brings the element of risk, the excitement that you may win or lose, the fun.

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