A bit lost on a game theory problem

[Ignaro]

Hello. First of all, forgive me if I post on the wrong category, I didn't find a better one.

There's a real life problem that I'd like to understand using game theory, but I have no formation on it and I am reading some introductory articles that aren't completely helpful, as my problem isn't quite featured on what I have read so far.

My problem is of openness and closeness of information. Say, as an example, two people have to be honest on whether one finds the other ugly.

If player A is honest and says B is ugly, B gains information, and A loses reputation. So, my first problem is, I have not seen an exemplified game where there's shared scores for the move of a single player, and don't know if it can be represented as such.

My hypothetical score is, if A is honest, 2 for B and -1 for A. 2 is chosen because if both are honest, the result for a single player would be 2-1=1, which is better than mutual closure, of gain 0 for each.

This is an image with my guess at the representation.


The second problem is, is this the best representation of the problem, considering players can lie and pretend they are being honest while not giving information damning for themselves? I'm honestly confused in this part.
My game seems to be imperfect information, non-cooperative and variable sum.

 

[Cubist]

I think the "ugly" and "honest" words distract me, and they don't seem relevant to the actual "core game" unless I'm missing something (which is often the case!). Please write back if I've missed something.

Would the words "gamble" and "hold" be possible alternatives? This is how I would interpret the same core game rules. Two players A and B go head to head at one time. Each player must choose whether to "gamble" or "hold" (without having knowledge of the other's choice while they decide). Choices are revealed simultaneously, and are scored...
If both players A & B gamble then A=A+1 and B=B+1
If only player A gambles, then A=A-1 and B=B+2
If only player B gambles, then A=A+2 and B=B-1
If no player gambles, then scores remain unchanged

 

[Ignaro]

I was trying to find an allegory without going into details, sorry for the confusion. The
closer to truth situation is that there's two players withholding information, and that each player profits from the other's honesty, while the honest player also MAY get a disadvantage, which I simplified for always.

Your model of gamble and hold seems correct, and if I'm not mistaken, gives the same table that I have attached. Given your answer I suppose this representation is correct for the simplified given premises.

But considering that (1) the negative impact is not a certainty and that, I now remember to add, (2) the order of play also plays a part (if A gambles then B has no reason not to hold) and that (3) both players may not know if the other is lying or not even after receiving information*, is that the best representation? I'm aware of trees, just don't know how to represent this situation.

*If A receives information negative to B, then he may assume information is true and complete. But if he receives information neutral to B, then it can be either true or false, or worse still, incomplete (which is a real life variable I think best not to introduce so far). So, it seems, if information is negative, then information is true.

I hope to have clarified it, although I realize I probably did the opposite! If you want more clarification, just say. Thank you!

 

[Cubist]

Your model of gamble and hold seems correct, and if I'm not mistaken, gives the same table that I have attached. Given your answer I suppose this representation is correct for the simplified given premises.

Indeed, yes, I think it gives exactly the same table.

When I think mathematically, I try to distill a problem down into a condensed form (leaving behind any words that don't affect the actual problem.) That's what I'm trying to do in order to understand your question.

there's two players withholding information, and that each player profits from the other's honesty, while the honest player also MAY get a disadvantage, which I simplified for always.

I think there are two problems with the honesty/ deceipt side of your game (which could be fixed with some thought)...

1) In your game the information being withheld doesn't seem concrete. There is no way for someone else to tell if player A thinks player B is ugly. Beauty is in the eye of the beholder. Everyone observing the game will have a different opinion about player B's "ugliness". But in poker, for example, the information being lied about IS concrete, the player can't change the value of the cards in their hand - everyone around the table is able to check the true worth of the card values (if everyone decides to call and thus the cards go face-up).

2) whether or not player A lies, AND assuming that we can work out if they are lying somehow, this doesn't affect the score. Let's say that player A says out loud "I think B is ugly" and player B says "player A is ugly". The end score is 1 for both players. How do we know if player A really thinks B is ugly? Player A could be telling the truth, OR they could be lying. But either way, the scores remain the same.

--------

As it stands, your game reminds me of an interesting UK TV quiz called "Golden Balls". Here's a video that explains the rules and shows a player who had an interesting tactic Golden Balls interesting tactic (click). The Wiki page on "Golden Balls" indicates this is similar to the Prisoner's dilemma

 

[Ignaro]

Yes, the example I gave was counterproductive.

About the problems you raised, the true game has concrete information, in the sense of being facts, as it refers to something the other player did, reasoned or believed - they don't change and the acts of reasoning and believing in a certain way/thing aren't interpretative, even if their contents are - although they are not verifiable.

So 1 doesn't seem to be a problem, but 2 does.

I don't see how to fix the verifiability of the game. The most players can do is to play considering behavioral game variables, such as appeal to the opponent's character. Although I don't have the knowledge to fully explore it, I see how belief, social norms and previous behaviour could affect this game in particular.

One interesting thing I found on the article you linked is that "There is little evidence that contestants’ propensity to cooperate depends positively on the likelihood that their opponent will cooperate", making the behavioral strategy, not rational in the purely mathematical sense, of starting the game providing honest information not seem so effective.

Here's a video that explains the rules and shows a player who had an interesting tactic Golden Balls interesting tactic (click)

Beautiful move from that player. He changed the game the moment he assured what he would do. With some more time to see that he was garanteeing not to receive steal from player 2, player 2 could have risked to steal, if he was so inclined to trust p1. Which kind that leads to the original dilemma, when without communication. At least, that's what I see.

Truly beautiful to see this kind of move, reminds me of when I used to play chess.[MATH][/MATH]

 

[Ignaro]

I consider the original question answered, thank you for the answers and the discussion. If you have more to contribute and wish to, I'm all ears.

I realise it is now veering into behavioral game theory territory, that I know nothing about! (Not to say the struggle to quantify (multiple) qualitative outcomes, that I noticed is more complex than my initial belief, and possibly related to methodology). Quite interesting, but probably beyond the scope of this forum.

 

[ellisael]

Wow this was really helpful as a thread esp the extra reading resources like Prisoner's dilemma etc.