## Pay to Win - a game theory problem

For the discussion of math. Duh.

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DrZiro
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### Pay to Win - a game theory problem

This is not homework; it's a problem I've been thinking about and can't quite figure out. As such, it may not be entirely well defined. I hope it's clear enough, otherwise I'll have to clarify.

Each player, A and B, is given a piece of paper with an amount of money written on it, which they read without showing it to the other player. We call their amounts a and b. They can then talk freely, offer each other money, and finally decide to defect or cooperate.

If only one decides to cooperate, that player gets nothing, and the other gets their number in money. For example, if A cooperates, A gets 0, and B gets b. If both defect, whoever has the highest number gets that number minus the other number. For example, if a > b, A gains a-b, and B gains 0. If both cooperate, they get nothing, so that's not really an option. Obviously their total gains are maximised if whoever has the lowest number cooperates. Now the question is, can they agree on some way of sharing the profit?

We assume that promises are kept; if one player says "I will give you this much if you cooperate", they can't go back on that.

If both numbers had been known to both players, we could reasonably argue that A stands to gain b by B cooperating, so that's the money they should split; thus, A should pay B b/2. But then B clearly has a reason to lie about the amount, and claim a higher number, let's say b'. Is there a way around it?

Demki
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### Re: Pay to Win - a game theory problem

If they can talk freely, won't they just produce a contract so that one player, say A, would give all his money to B, then A cooperates and B defects, and they would share the money 50-50?
Sure B could lie about the money, but in the contract A can demand to see the piece of paper of B

DrZiro
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### Re: Pay to Win - a game theory problem

Okay, let's say they're only told an amount, rather than given a piece of paper.

Demki
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### Re: Pay to Win - a game theory problem

I'd say that in that case, the contract will be such that whatever money B gives to A, A returns that amount after A defects and B cooperates, since neither side can know the other's amount. Without a contract, A could always run off with the money. This way each side gains the money that was told to them, which is more than getting 0.

quantropy
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### Re: Pay to Win - a game theory problem

In experiments where people have to split money between them, there tends to be a strong feeling that it should be split evenly. A player would prefer both getting nothing to what is seen as an unfair split.

I've a feeling that an even split of the final amount is probably the only workable solution in this case. A defects, B cooperates and they get a/2 each.

If the contract has to mention specific amounts of money rather than agreeing to split it 50/50 then there may still be an incentive to lie, but I'm not sure how to deal with that.

PeteP
What the peck?
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### Re: Pay to Win - a game theory problem

Naming a lower number carries a risk though, namely that the other one might have more than your false one => you end up getting half of theirs.

Demki
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### Re: Pay to Win - a game theory problem

Ah, I think I have been assuming something not given, that they can transfer the money they were given BEFORE they choose to defect/cooperate. I guess I though too much about Liar Game(a psychological manga that has plenty of interesting games)

DrZiro
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### Re: Pay to Win - a game theory problem

I don't see how it would matter whether the money is transferred before or after?

Quantropy: The way I thought of it, both defecting is the default state, so their perceived fair solution is splitting the gains above that. But the other way is also possible. I guess it depends on how you express the game. Either way, they don't know how much that is. They won't be able to see how much money the other person is getting afterwards either.

Demki
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### Re: Pay to Win - a game theory problem

Well, if they can transfer before, then B could give A all the money he "got", then cooperate while A defects, giving A a+b while b still gets 0(and doesn't lose any because he only transfered the money that was allocated to him)

DrZiro
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### Re: Pay to Win - a game theory problem

He could, but why would he? He doesn't know that A got more, so it's in his interest to defect.

Demki
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### Re: Pay to Win - a game theory problem

DrZiro wrote:He could, but why would he? He doesn't know that A got more, so it's in his interest to defect.

Say A got a money and B got b money.
Should they produce a binding contract such that they collect all the money into one player's "bank", then they "cash out" a+b money and return each person's money to him(if they collect in A's bank, A gives B b money after he gets the cashed out a+b money, and vice versa), then A gets a money and B gets b money.
Should they both defect:
If a>b then A gets a-b money which is less than a, and B gets 0 money which is less than b.
If b>a then B gets b-a money which is less than b, and A gets 0 money which is less than a.
If b=a, they both get 0 money I guess.
So they both stand to gain money by having all the money in one player's hand and defecting, then sharing the sum a+b such that A gets a and B gets b.

But that only works if they are allowed to do that, which I mistakenly assumed, as it wasn't written in the OP.

Tirear
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### Re: Pay to Win - a game theory problem

Both players write down a number, and hand it to the other player at the same time. The one who wrote the lower number cooperates, and the other gives him half of that number in dollars.
Suppose one player is considering writing a number higher than his real number. If the new number still ends up less than what the other player writes, he gets extra money. If the deception causes his number to be higher than what the other player writes, he gets less money. If his real number is greater than what the other player writes, the deception has no effect.
I notice that a backfiring deception always costs less profit than it would grant if it had been successful, however I feel that the lack of knowledge of how much deception one could expect to get away with combined with the typical nonlinear value of money should result in little or no deception. EDIT: Then again, I never was one for casinos. Typical people may be fine with cheating themselves out of money, especially considering the other player is also worse off.
DrZiro wrote:Quantropy: The way I thought of it, both defecting is the default state, so their perceived fair solution is splitting the gains above that. But the other way is also possible. I guess it depends on how you express the game.

Double defect has to be the standard for comparison, because it is the worst you have to threaten your opponent with. If you have 10 and the other player has 100, and you demand that he promises you \$50 or you'll ruin everything by cooperating, he just defects and accepts his \$100. A double cooperate would require his cooperation (does this count as a pun?). Simply expressing the game differently can't change that, you would have to change the rules.

DrZiro
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### Re: Pay to Win - a game theory problem

Demki wrote:Should they produce a binding contract such that they collect all the money [---]
But that only works if they are allowed to do that, which I mistakenly assumed, as it wasn't written in the OP.

Right, they can't do that, because neither knows how much "all the money" is. It could have been clearer in the OP.

Demki
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### Re: Pay to Win - a game theory problem

DrZiro wrote:
Demki wrote:Should they produce a binding contract such that they collect all the money [---]
But that only works if they are allowed to do that, which I mistakenly assumed, as it wasn't written in the OP.

Right, they can't do that, because neither knows how much "all the money" is. It could have been clearer in the OP.

Actually that's not what I meant by if they can do that. If they can transfer money, then all the contract has to say 'B transfers b money to A, A defects and B cooperates, A returns b money to B'. Now in this contract B has no incentive to lie about his money, since if he states a lower amount he loses money, if he states higher he won't be able to transfer in the first place.

DrZiro
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### Re: Pay to Win - a game theory problem

Maybe I should rephrase that part. That's a good thing about posting here - even if you don't find the answer, you find out how to clarify the problem.

So I'll replace
We assume that promises are kept; if one player says "I will give you this much if you cooperate", they can't go back on that.

with
We assume the players carry enough money in their wallets to pay each other for any deal they come up with. If a player agrees to cooperate, we also assume they won't go back on that promise.

A deal can only consist of giving a specific sum and getting cooperation. You can't promise to show how much you're getting, or promise to pool the money afterwards.

SDK
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### Re: Pay to Win - a game theory problem

Tirear wrote:Double defect has to be the standard for comparison, because it is the worst you have to threaten your opponent with. If you have 10 and the other player has 100, and you demand that he promises you \$50 or you'll ruin everything by cooperating, he just defects and accepts his \$100.

He doesn't get \$100 for defecting, he gets \$90 (his \$100 minus your \$10 - because you would choose to defect here too if he didn't want to go with your sharing plan). If you lied and said you had \$75, he'd think defecting would only gain him \$25, so it appears to be in his best interest to go along with your plan of splitting his \$100 fifty-fifty (after you cooperate and he defects - the only way he can get the full value). If they both told the truth, the guy with \$10 can only really ask for \$10, and that's if the \$100 guy is feeling generous since he gains nothing from netting you that \$10 (he gets \$90 either way, so I guess you'd better ask for \$9 instead).

So it appears that inflating your number is the way to go, but only if you have the lower number. If you have the higher number (something you are not aware of to start), inflating your number just gives the other player more bargaining power. Lowering your number doesn't do anything though, even then, because if \$10 guy thinks you only have \$50 (when you really have \$100), he can still only ask for \$10, and you still get away with \$90, same as you would have otherwise. Oh, actually, that applies to inflating your number too (if you say you have \$200, he asks for \$10, same as ever). Based on that, in this situation, you should always inflate your number.

But that's only for numbers that are far enough apart. If you've got \$90 and I've got \$100, defecting is only going to gain me \$10. Cooperation (ie: one of us defecting while the other cooperates) is going to net us a big jump in profits. This is especially true if our numbers are the same (both \$100, say), so that coming to an agreement is the only way either of us are going to profit. If this is the case, lying about your number is going to seriously hurt you. If I say I have \$200 and you (truthfully) say you have \$100, you're not likely to settle for much less than \$99. I have to give up 99% of my profits now? Ouch. Even if I can convince you that the difference is going to be mine anyway (since I can just walk away with \$200 minus \$100, that \$100 is already mine), and therefore we should share the extra \$100 evenly, we're still just making the same even split we would have made if I'd told the truth (\$50 each - which is what we actually get). So... I guess we're back at lying being equivalent at worst.

I'd have to run more numbers to be sure, but it's hard to see a downside to lying. Most of the time it won't gain you anything (while also not losing you anything), but sometimes it will gain you quite a bit. I suppose it depends on who your negotiating with. A lot of people I know would never make a deal where I get away with \$150 to their \$50, even if the \$100 is already guaranteed for me.
The biggest number (63 quintillion googols in debt)

Cauchy
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### Re: Pay to Win - a game theory problem

Once you open up lying, then why should I believe your inflated number? If I know you're liable to inflate your number, then I have no reason to assume that the number you report is correct.
(∫|p|2)(∫|q|2) ≥ (∫|pq|)2
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Tirear
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### Re: Pay to Win - a game theory problem

SDK wrote:
Tirear wrote:Double defect has to be the standard for comparison, because it is the worst you have to threaten your opponent with. If you have 10 and the other player has 100, and you demand that he promises you \$50 or you'll ruin everything by cooperating, he just defects and accepts his \$100.

He doesn't get \$100 for defecting, he gets \$90 (his \$100 minus your \$10 - because you would choose to defect here too if he didn't want to go with your sharing plan).

If you re-read the part you quoted, I was talking about a player threatening to cooperate. That doing so would be immensely stupid was the entire point.
SDK wrote:So... I guess we're back at lying being equivalent at worst.

I'd have to run more numbers to be sure, but it's hard to see a downside to lying. Most of the time it won't gain you anything (while also not losing you anything), but sometimes it will gain you quite a bit.

Inflating your number hurts you whenever it causes you to have the higher number. If you have 100 and say 100, and I have 10 but say 150, my lie costs me \$45 (most of which is coming from my own pocket). If you have 100 but say 200, and I have 10 but say 150, your lie costs you \$25.

Tyndmyr
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### Re: Pay to Win - a game theory problem

DrZiro wrote:If only one decides to cooperate, that player gets nothing, and the other gets their number in money. For example, if A cooperates, A gets 0, and B gets b. If both defect, whoever has the highest number gets that number minus the other number. For example, if a > b, A gains a-b, and B gains 0. If both cooperate, they get nothing, so that's not really an option. Obviously their total gains are maximised if whoever has the lowest number cooperates. Now the question is, can they agree on some way of sharing the profit?

God no.

Look, make a grid up, which shows my cooperation/defection vs my opponents cooperation/defection. There is literally no reason for me to cooperate, ever, because if I do, either case results in me getting nothing.

If I defect, maybe my opponent will stupidly cooperate, or maybe I'll have the higher number.

Talking is futile in this situation, since I'm mashing the "defect" button regardless.

(Yes, I'm aware that I can apparently ask for a bribe for helping my opponent to win more. This should only ever be done if you are absolutely confident that you have the smaller number, and have no other path to gain anything)

Layco
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### Re: Pay to Win - a game theory problem

Seems like a simple enough Game.

There is no Nash Equilibrium so without making promises (to get out of the standard matrix of outcomes) there will be no way for them to agree to any set of moves.

But with promises, there is a Nash Equilibrium. The player with higher endowment (for this post it will be player A; if it is B then just switch the notation) will make an offer to B. "I will give you 1 unit of money if you cooperate while I defect".

If B doesn't cooperate under these terms, then he can either cooperate without these terms (and get 0, given that A will defect), or he can defect under these terms (and get 0, as he has the lower endowment).

edit: It seems that you guys are assuming that player B would rather have 0 units of money than 1 unit of money. I don't recall anyone presenting a social-norms or a fairness function, so we can't take that into account.

PeteP
What the peck?
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### Re: Pay to Win - a game theory problem

By the same reasoning B can say they will cooperate if A gives them b-1 since with their cooperation A gets 1 coin more so they should do it. (Ignoring the option of lying about the amounts of a and b.)

Layco
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### Re: Pay to Win - a game theory problem

That's true PeteP - A gets (a-b+1) and B gets (b-1) if a>b and if b>a then you just reverse the notation.