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Let me start over. I misread the payouts.

p11 is {Stag,stag}=4
p00 is {Hare,hare}=3
p01 is {hare,stag}=0
p10 is {stag,hare}=0

Using the law of excluded middle
Pstag + ~Pstag = 1
~Pstag = 1 - Pstag

Let Pstag be represented by x, and the yield by f(x).

f(x)=4*x^2+3*x*(1-x)+3*(1-x)*x
f(x)=4*x^2+6*x*(1-x)
f(x)=4*x^2+6*x-6*x^2
f(x)=2*x*(x+3-3*x)

Next we find the second derivative and solve for 0. Am I making any mistakes so far?

This assumption is quite a big one -- why should we assume this? Standard game theory (including cooperative game theory) doesn't assume that individuals are aiming at maximizing for the group -- if it did, the prisoner's dilemma wouldn't be much of a dilemma.

It is not a single maximizing function that matters -- it is a pair of maximizing functions, you need to look at the payoffs/actions for both players simultaneously...

What you're looking for in your last two posts, I believe, is the formula for calculating the payoffs, right? If so lets say x is the frequency that player 1 plays stag, and y is the frequency for player 2 playing stag.

Payoff to player 1:
4xy + 0x(1-y) + 3(1-x)y + 3(1-x)(1-y) -- which reduces to...
4xy + 3(1-x) -- call this u(x,y) for the utility function when player 1 plays stag with probability x and player 2 plays stag with probability y

Payoff to player 2 is the same, but with x's and y's reversed.

What we want to do is to look at u(x,y), for player 1, and u(y,x), for player 2, to determine the nash equilibria -- find the values of x and y such that neither player can do better by changing. This is similar to working with the derivatives, but in this game the solutions are very easy to find...

There are 3 Nash Equilibria (i.e. solutions to the game)
1. x=1, y=1 (both players choose to hunt stag, i.e. stay with probability 1)
2. x=0, y=0 (both players choose to hunt hare, i.e. go with probability 1)
3. x=0.75, y=0.75 (both players choose to hunt stag with probability 0.75)

In each of these cases neither player can improve by doing choosing an x (or y) that is different.

Equilibria 1 and 2 are the most intuitive equilibria. In the first one, both players choose to stay at their post hunting stag and of course neither would like to deviate because if they did they would get a payoff of 3 rather than 4. In the second one, both players choose to hunt hare (i.e. go) and neither would like to switch because they would get nothing with the other player hunting hare.

Solution #1 is called the payoff dominant equilibrium (because both players get 4, the best they can possibly do). Solution #2 is called the risk dominant equilibirum because that is the safe option if the players are in doubt about what there opponent will do (i.e. if they guess that there opponent is about 50-50 likely to play each strategy).

Solution #3 is a mixed-strategy equilibrium. If your opponent chooses to hunt stag 0.75 of the time, it doesn't matter what you choose, you are expected to get exactly the same payoff. If you choose the same, he is in the same circumstance so neither of you can do better by choosing something else, so it is a nash equilibrium. This solution is a bit strange however, since it requires both players choosing a very precise probability -- this is said to be an "unstable" solution.

Yes, that's what I was trying to find, Nash equilibrium. I need to re-read. It's been a couple years.

There are several reasons that the Stag Hunt is interesting, to game-theorists and philosophers alike.

First, since there are multiple Nash Equilibria of the game, we are left with the question: what does game theory predict (or recommend) of people faced with this game? There is no easy answer to this question -- there is a good justification for playing both strategies.

Second, the game is taken to be the simplest representation of what philosophers call a "social contract". If both players can manage to find the cooperative high-payoff equilibrium, this is something that is stable (and both benefit from), but it is risky and you only want to do it if you know the other one will also do it. If we can explain how cooperation can arise in the Stag Hunt, we can start to explain how social contracts can arise.

What I'd do is propose trade. We alternate, so we both maximize. Of course, no communication is possible is the general assumption. There are infinite strategies possible, but most significant to me are these. Here are some mixed strategies that aren't random, but adopt a list.

S1={stag,hare,stag,hare,stag,hare,...}

S2={hare,stag,hare,stag,hare,stag,...}

With little imagination, both can figure out that what is best for each of them as individuals is for A to adopt S1 and B to adopt S2.

My goal would be for this to happen as its a win/win. After adopting either S1 or S2, my partner is either rewarded for adopting a strategy that enjoys perfect parity, or suffer by adopting something inferior. In this way, even with out actual communication (although, I train my partner to behave in theory), we both land on the same page.

However, I could draw someone who stubbornly does S3. He could be described as a cross between a gamesman and a fair-man. Perhaps a less-than-fair-man?

Here are two more interesting strategies.

S3={stag,stag,stag,hare,stag,stag,stag,hare,...}
S4={hare,hare,hare,stag,hare,hare,hare,stag,...}
Then, I can either counter as my fair-man routine of alternating with S2, or respond as a 'rational' which results in the quickest and most profitable in the very short-term with S4.

Although in the short-term it appears that S4 would be the most rational thing, I would think it would still be probable that after adopting S1, or S2, my partner would eventually adopt the otherand I'd simply stick with it and expect a long term return on my diplomatic efforts.

You're thinking the game is more complicated than it is intended to be... on the one hand there is no repetition of the game (hence no trading off that can be done). On the other hand, it is trivially easy to identify what the optimal outcome is for both: when each of them hunt stag (i.e. stay at their post).

You can grant that they both know what the game is, that both would choose the (stag,stag) outcome if they could pick any outcome they wanted, and that both know the other would really like to hunt stag -- the problem is one of trust. How do I know the other guy will do his share (it is in his interest to do so, but only so long as he expects that I do my part -- but how does he know this? etc)

Lets focus on the two pure-strategy equilibria of the game for a moment -- (i) where both players hunt Stag and (ii) where both players hunt Hare. The question could be put as follows: which equilibrium should I try to play? There are good arguments for both and the problem is not that cooperating in this game (hunting stag) is not rationally justifiable, it is. The problem is that either strategy is rationally justifiable, so how do you choose one? Here are the arguments for each (i) and (ii):

(i) I will play stag in an attempt to reach the nice (stag,stag) equilibrium because that is the optimal equilibrium -- and my partner agrees this is the optimal equilibrium.

(ii) But, I can't be guaranteed that my partner will trust me -- he might doubt my cooperation. If there is a good chance that he'll play hare, I should do so as well (otherwise I will get nothing). So, hunting hare is the risk-dominant (i.e. less risky) equilibrium.

I got the payoffs mixed up again.
(stag,stag) yields 4
(hare,stag) yields 3
(stag,hare) yields 0
(hare,hare) yields 3

Besides the complex mathematically equations, I suppose what I was trying to say that trust builds over time. It's not a stable game for all the reasons PR pointed out. I did not realize it is only played once.

OK here is a game that raises many of the same issues as the Stag Hunt but some people find it a much more compelling game to think about. I will just post the game, not the game theoretic "solutions" yet because I'm interested in learning what people would actually do in this game.


Battle of the Sexes:

A pair of newly weds visiting the big city get separated right after having agreed to go to EITHER the baseball game that evening or the opera that evening (they have tickets to both and each is holding their own ticket) -- but they hadn't settled on which event they were going to go to (and they don't have cell phones or any way of contacting the other person).

The husband (H) would prefer to go to the baseball game over the opera, but only if he is sure meet up with his wife

The wife (W) would prefer to go to the opera over the baseball game, but only if she is sure to meet up with her husband.

This situation is captured in the following payoffs:

For H:
2 if both H and W go to the baseball game
1 if both H and W go to the opera
0 otherwise

For W:
1 if both H and W go to the baseball game
2 if both H and W go to the opera
0 otherwise

Both know and understand the situation, and both knows the preferences of the other. What would you do if you were in the position of H? W?

I have a question here.

Why is the payoff 0 even if they go to the event they wanted to go to more, if they don't meet up with the other person?

Example: If H went to the Opera and didn't meet up with his Wife, woudln't that be worse than going to the Baseball Game and not meeting his wife? At least he would have gotten to see the game.

On the other hand... he'd get in trouble for not going to the Opera. I suppose that makes sense.

I guess as H, he better go to the Opera. Even if he doesn't meet his wife there, he can't get in trouble for it. If he goes the Baseball game and she isn't there, man he's sleeping on the couch for his honey moon. XD

The 0 captures the idea that if they miss-coordinate H won't enjoy himself no matter where he is. Think of it as this: if he isn't with his wife he will be so upset it won't matter whether he is at the opera or the game. Same is true for W.

As far as what H should do -- well, he should only go to the Opera if he expects W to go to the opera, because he will wind up "sleeping on the couch" anyway if he doesn't coordinate.

Also, W could go through exactly the same reasoning (since her payoffs are symmetric and wind up going to the baseball game). If they both try to go to the other person's preferred place, they wind up missing eachother and both end up terribly upset.

The 0 captures the idea that if they miss-coordinate H won't enjoy himself no matter where he is. Think of it as this: if he isn't with his wife he will be so upset it won't matter whether he is at the opera or the game. Same is true for W.

As far as what H should do -- well, he should only go to the Opera if he expects W to go to the opera, because he will wind up "sleeping on the couch" anyway if he doesn't coordinate.

Also, W could go through exactly the same reasoning (since her payoffs are symmetric and wind up going to the baseball game). If they both try to go to the other person's preferred place, they wind up missing eachother and both end up terribly upset.

I would argue that you are mistaken. I beleive that H should go to the Opera, because if he doesn't, he will be in big trouble. I also beleive that W will go to the Opera, because she knows she will meet H there, because if she doesn't meet him there, he is in big trouble. =D

So gender is the symmetry-breaker in the situation. When in doubt, side with the female preferences

Supposing H and W are both lesbians... now what?

The solution is the opera. The symmetry is broken in the situation by the intellectual and emotional refinement and depth of the one option against the hedonic triviality of the other. If the husband desires to fritter his time away on baseball while he has tickets to an opera, that is a vice, and so even if his wife were not a part of the equation, he would do himself a favour by overcoming himself and going to the opera.

All bombastic joking aside (I am just excited because I just went to the opera the other day), I think that a problem in the game is actually the fact that we are weighing the pleasure of the husband and wife, even if it is more than merely the pleasure of the event (being that it is the pleasure of social intimacy which is rewarding). If individual personal growth were the thing measured instead, then my half-sarcastic paragraph above might actually be possible to take somewhat seriously.

You raise a very deep interesting question Brian -- is it possible to compare one person's pleasure (happiness, well-being, whatever) with another person's in some objective way? It seems hard to imagine that we could really do this in an objective way, but intuitively we make such comparisons all the time (we say things such as "I would like that last piece of cake, but I know you like it more, so you have it" and it seems like these things have some coherent meaning...)

Anyway, although this point is a very good one, we don't actually require such interpersonal comparisons in order to think about the game -- perhaps if we wanted to choose an outcome as some sort of social dictator, we would, but from the perspective of the players in the game the specific values of the payoffs to the other player are not supposed to matter.

What do I mean by this? Well, the idea is that the payoffs of player 1 (or player 2) capture EVERYTHING that player might care about (including the possible well-being of the other players). So if one outcome gets a 2 and another gets a 1, that means they prefer the outcome that gets 2 to the outcome that gets 1 all things considered (because the number, by assumption, has all things considered). Think of the payoff as the overall resulting mental state of each player. It becomes almost tautological (it actually is tautological on the mainstream theories) that players will aim to maximize this number -- but maximizing is not some sort of blind self-interest, because players may care about others as well -- it is just trying to get what you want to get (that is the tautological part).

So, from the perspective of each players WITHIN this game, they don't need to do any "weighing" their payoff against the payoff of the others (if this affects their well-being, it is already factored in). The strategic situation that they are faced with is there without the need to think about this sort of "weighing" going on (the need only arises if we want to assess different outcomes objectively OUTSIDE the game, but the players are not trying to do this). What I'm trying to say is that the strategic problem in battle of the sexes is formulated (or could be formulated) so that any "weighing" going on is already factored in and so is irrelevant to finding a solution.