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Consider the following version of the ultimatum bargaining game: Player 1 has the first move. He can choose how to divide a cake of size normalised to 1 and offer player 2 a part of this cake, say x, which is an element of the closed interval [0,1]. Player 2 gets to know player 1's offer and can accept the offer or reject it. If he accepts, he gets x and player 1 gets 1-x. If he rejects, both player get nothing.

My questions:
1. What is the Nash equilibrium of this game?
2. Is this Nash equilibrium subgame perfect*? Why (not)?

*A Nash equilibrium for an extensive form game (an extensive form with perfect information is basically a rooted tree with a partition of the set of moves) is subgame perfect if it induces a Nash equilibrium in every subgame.
Henry Swift (guest) meinte am 19. Oct, 23:07:
Hmmm
There are actually two Nash equilibria, one in which P1 gives nothing to P2 and P2 accepts the split and one in which P1 gives nothing to P2 and P2 rejects the split. They are both subgame perfect. 
Henry Swift (guest) antwortete am 19. Oct, 23:28:
Oh, and...
There are also uncountably infinite mixed equilibria that all include P1 giving nothing to P2 and P2 accepting with any real probability p and rejecting with probability 1-p. These are also subgame perfect. 
Dr Ecksau (guest) antwortete am 20. Oct, 01:53:
Why should
{offer zero, reject offer} be a Nash equilibrium? 
Henry Swift (guest) antwortete am 20. Oct, 12:07:
If P1 offers 0, then P2 can't improve the result from accepting. If P2 rejects, P1 can't improve the result from offering more than 0. 
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Johan Richter (guest) meinte am 20. Oct, 00:01:
Isn't there more equilibrias?
Is not the following a Nash equillibrium?

Player 1: Offer x=0.1

Player: Accept if x=0.1. Reject otherwise, even if x is larger than 0.1.

Given these strategies it does not pay for player 1 to offer anything else since in that case he would get nothing. It does not matter what strategy player 2 has when x<> 0.1 since it never happens so he can keep the current strategy.

Of course it is not sub-game perfect. 
MephistoS antwortete am 20. Oct, 00:49:
Little hint concerning subgame perfection
Don't forget that we have an infinite strategy space. This makes subgame perfection tricky. 
Henry Swift (guest) antwortete am 20. Oct, 12:09:
Right...there are more I hadn't considered. Lots more. 
jason049 (guest) antwortete am 24. Aug, 12:18:
Don't forget that we have an infinite strategy space. This makes subgame perfection tricky. GE.TT 
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Brock (guest) meinte am 20. Oct, 02:28:
Math isn't me thing, but ...
I do know a little bit about this game. The results seem (to me) so dependant on the psychology of the offeree that the offeror's strategy must depend on his knowledge of the psychology. Can Nash's formula take this into account? I honestly don't know, since as I said, math isn't my thing.

As an anecdote, I participated in this very game recently. I turned down $4 out of $10. Maybe if my partner had know I require fairness in all my dealings his strategy would have been different. Other people who aren't as strict as I am took as little as $1. Different psychology. 
Mahalanobis antwortete am 20. Oct, 04:14:
The Economics of Fair Play
"Recently an ambitious cross-cultural study in 15 small-scale societies on four continents showed that there were, after all, sizable differences in the way some people play the Ultimatum Game. Within the Machiguenga tribe in the Amazon, the mean offer was considerably lower than in typical Western-type civilizations—26 instead of 45 percent. Conversely, many members of the Au tribe in Papua New Guinea offered more than half the pie. Cultural traditions in gift giving, and the strong obligations that result from accepting a gift, play a major role among some tribes, such as the Au. Indeed, the Au tended to reject excessively generous offers as well as miserly ones. Yet despite these cultural variations, the outcome was always far from what rational analysis would dictate for selfish players. In striking contrast to what selfish income maximizers ought to do, most people all over the world place a high value on fair outcomes. <>

Getting Emotional: Economists have explored a lot of variations of the Ultimatum Game to find what causes the emotional behavior it elicits. If, for instance, the proposer is chosen not by a flip of a coin but by better performance on a quiz, then offers are routinely a bit lower and get accepted more easily—the inequality is felt to be justified. If the proposer’s offer is chosen by a computer, responders are willing to accept considerably less money. And if several responders compete to become the one to accept a single proposer’s offer, the proposer can get away with offering a small amount.

These variations all point to one conclusion: in pairwise encounters, we do
not adopt a purely self-centered viewpoint but take account of our co-player’s
outlook. We are not interested solely in our own payoff but compare ourselves with the other party and demand fair play.

Why do we place such a high value on fairness that we reject 20 percent of a
large sum solely because the co-player gets away with four times as much?
Opinions are divided. Some game theorists believe that subjects fail to grasp
that they will interact only once. Accordingly, the players see the offer, or its rejection, simply as the first stage of an incipient bargaining process. Haggling about one’s share of a resource must surely have been a recurrent theme for our ancestors. But can it be so hard to realize that the Ultimatum Game is a one-shot interaction? Evidence from several other games indicates that experimental subjects are cognitively well aware of the difference between oneshot and repeated encounters.

Others have explained our insistence on a fair division by citing the need, for
our ancestors, to be sheltered by a strong group. Groups of hunter-gatherers depended for survival on the skills and strengths of their members. It does not help to outcompete your rival to the point where you can no longer depend on him or her in your contests with other groups. But this argument can at best explain why proposers offer large amounts, not why responders reject low offers.

Two of us (Nowak and Sigmund) and Karen M. Page of the Institute for Advanced Study in Princeton, N.J., have recently studied an evolutionary model that suggests an answer: our emotional apparatus has been shaped by millions of years of living in small groups, where it is hard to keep secrets. Our emotions are thus not finely tuned to interactions occurring under strict anonymity. We expect that our friends, colleagues and neighbors will notice our decisions. If others know that I am content with a small share, they are likely to make me low offers; if I am known to become angry when facing a low offer and to reject the deal, others have an incentive to make me high offers. Consequently, evolution should have favored emotional responses to low offers. Because one-shot interactions were rare during human evolution, these emotions do not discriminate between one-shot and repeated interactions. This is probably an important reason why many of us respond emotionally to low offers in the Ultimatum Game. We may feel that we must reject a dismal offer in order to keep our self-esteem. From an evolutionary
viewpoint, this self-esteem is an internal device for acquiring a reputation,
which is beneficial in future encounters."

Source: The Economics of Fair Play, Karl Sigmund et al. 
Al (guest) antwortete am 20. Oct, 05:58:
What if it was over a 10k pie
Brock,
You may have turned down $4, but would you have turned done $4,000 over a sense of fairness? Probably not. 
Brock (guest) antwortete am 20. Oct, 18:14:
If I were you, I wouldn't bet $6,000 on that.

When I say I require fairness, I mean it. Size matters not. 
Tony Candido (guest) meinte am 20. Oct, 06:26:
I'll bite
1. Infinite Nash equilibria between x=epsilon and x=1 (at x=0 P2 has no incentive to accept).

2. One of them is subgame perfect: x=epsilon. That is the rational maximizing strategy for P1, assuming P2 is rational, and P2 knows it. 
Henry Swift (guest) antwortete am 20. Oct, 12:13:
At x=0 P2 has no incentive to accept, but that's not the definition of a Nash equilibrium. A Nash equilibrium is a pair of strategies such that neither player can do better given the other player's strategy. If P1 offers 0, P2 can't do better than accepting (or rejecting, for that matter.) And x=epsilon can't be part of a Nash equilibria since it is always better for P1 to offer x=epsilon/2. 
Paul N (guest) meinte am 20. Oct, 10:07:
I've been a subject for a few experimental economics experiments (back when $30 for 2 hours seemed worth it), and I always played rationally (e.g. accept any offer in a single-trial ultimatum), but when I'm on the bum end of games, it makes me feel yucky and I never want to be a test subject again. There was this one where the best strategy was clear to everyone, and player type A got 0.75 average per round, and player type B got 0.25 average, and I was so pissed about being a B I wanted to walk out of the experiment just to fuck it up.

What are really fun are the trading games where they try to see how efficient the (fake) market is or how fast you react to new information or conditions; second most fun are games where the faster you go, the more $ you get. It feels so good to see your ID at the top of the payout list.

The other reason I stopped enjoying these experiments is that the equilibria / optimal strategies were usually so *obvious* that soon everyone followed them and playing wasn't interesting. I could just see the paper the grad student was planning to write based on the experiment, and I imagined them smugly concluding that their findings were novel and significant when they were actually really boring and irrelevant. Plus, I never understood why you could write papers that were supposed to represent the behavior of humans as a whole, when your sample set was a bunch of IQ>130 students, most of whom knew at least a bit of game theory. 
MephistoS meinte am 20. Oct, 13:06:
New Record of comments :D
Hi to all again, thx for your great comments.
It seems that this game is well known to many readers.

When I formulated my questions I didn't really think about multiple Nash equilibria. I had this one equilibrium in mind many of you pointed out- that x=0 and player two accepts any offer greater or equal to 0.

One of you decribed this nice experiment where he actually played (almost) subgame perfect if the strategy space of player 1 were finite.
But player 1's strategy space in an intervall, hence a continuum.

So the for me striking result is that the above game doesn't have a subgame perfect equilibrium! Why?
(Hint apply Kuhn's lemma and its two critical conditions finiteness and the expected utility hypothesis which can be taken as given.


Solution:
Suppose x=0, accept x >=0 is subgame perfect.
Construct a sequence of player 1's offers which approach 0. What happens to the pay-off function of player 1 in the limit of this sequence? It's discontinuous in the limit! As indifference of player 2 to accept or reject the offer of x=0 can be interpretated as flipping a fair coin.
Weierstrass' Theorem doesn't work, right?

Next suppose player 1 offers x being close to 1 ==> His strategy space isn't compact, hence Weierstrass' Theorem doesn't work again. Player 1 cannot maximize on an open interval.

==> No subgame perfect equilibrium for this game! 
ulrich (guest) antwortete am 24. Oct, 14:34:
wrong...
Of course there is a subgame perfect equilibrium, and it is indeed (offer 0, accept all offers). First, this is obviously a Nash equilibrium. Second, in all the nontrivial subgames, player 2 can not do better than accept, so this strategy induces a Nash equilibrium in every subgame. That's it - Kuhn and Weierstrass are only distracting here, and player 1's strategy space is [0,1], which is clearly compact.
What you perhaps meant is that (offer 0, accept all positive offers and reject 0) is not subgame perfect. Well, it's not even a Nash equilibrium, since player 1's best response set is empty in this case... 
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MephistoS meinte am 24. Oct, 18:31:
My mistake
I should have stated that Indifference means that player 2 randomizes somehow between accepting or rejecting an offer of x=0,
where both events get a stictly positive probability.

Then no subgame perfect equilibrium exists. 
ulrich (guest) antwortete am 24. Oct, 18:57:
Existence of SPE...
... does not depend on your assumptions about how players do, would, or should behave. It is simply a property of the game, i.e. (among other things) of how players *can* behave. If the responder (for whatever reasons) throws a coin in case of indifference, then there still *exists* a SPE, it is just not played.
If you want to arrive at a game without a SPE, you have to change the rules of the Ultimatum game, e.g. by demanding that the responder *has to* mix (with strictly positive probabilities) in case of a zero offer. However, this is a different game, and also the restricting assumption would be rather ad hoc. 
MephistoS meinte am 24. Oct, 19:23:
hmm that's it
I wanted to force player 2 to mix , facting x=0. It's an ad hoc assumption. And I changed the rule how "indifference" is "handled". Thanks for your comments. I hope, now it's correct. 
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