User:Benja/Stay Firm or Give In



 Stay Firm or Give In  is a two-player, non-zero-sum, game-theoretic game I made up and am interested in studying. Two players take turns either 'staying firm' or 'giving in'. If one player gives in, the other player gets one more turn; if the other player stays firm, the payoff is +5 for the player who stood firm and -5 for the player who gave in, but if both players give in, the payoff is +10 for both players. If neither player gives in, the game ends after two turns with a payoff of (0,0).

As currently written, this page assumes familiarity with the basic concepts of game theory (but nothing beyond what's behind the Wikipedia links).

Update (2008-04-06): Yay! The point I'm getting at with this is known in the literature under the name of self-confirming equilibrium (Fudenberg and Levine (1993)); I've found it by way of w:Wikipedia: WikiProject Game theory. To do: Read the Econometria article, update this page, start the article at Wikipedia.




 * The article briefly discusses a game, shown on the right, that is subtly different from a one-round Stay Firm. The difference is that in their game, Public Rationalization, the best outcome for player 2 is if player 1 plays left ("stays firm"). I'm not sure yet whether this makes any substantive difference to analysis of the game.
 * Strangely, the authors state that there are two Nash equilibria, (L) and (R,U), which sounds as if they're saying both (L,D) and (L,U) are Nash; but in (L,U), player 1 receives a better payoff by switching to R. (L,U) is the missed opportunity outcome that makes me interested in Stay Firm, and it's "clearly" a self-confirming equilibrium that is not Nash, in either Public Rationalization or Stay Firm. The authors don't mention that; rather, their interest is in the mixed strategy profile ((1/2L,1/2R),U), which is a self-confirming equilibrium "whose outcome is a convex combination of the Nash outcomes." It looks like I'll have to find out what "convex" means in this context :-)
 * Co-author David K. Levine's book, The Theory of Learning in Games, gives the game in a figure with the caption, "Public randomization in a Selten game." (fig. 6.4, p.189)

Strategy profiles
I find it useful to divide the strategy profiles of this game into the following categories:


 * Happy ending profiles are those in which both players give in (payoff: +10/+10).
 * Tragic ending profiles are those in which one player gives in, but the other stays firm (payoff: -5/+5).
 * Missed opportunity profiles are those in which neither player actually does give in, but at least one player would have given in if the other had given in first (payoff: 0/0).
 * The butt-head profile is the profile in which neither player gives in or would have given in under any circumstances (payoff: 0/0).

Note that each strategy profile of Stay Firm or Give In fits in exactly one of these categories.

I'm particularly interested in the profile where both players would have been willing to give in at every turn, but neither actually does. Call this the canonical missed opportunity profile. I'm interested in this profile because it seems to me that there are many situations in real life that can be modelled in terms of this game in which both players at least claim to act as described by it.

Note that in the canonical missed opportunity profile, both players play the unique Minimax strategy available to them.

Equilibria and rationalizability
It's obvious that the happy ending profiles are Nash equilibria. It's also easy to see that the butt-head profile is a Nash equilibrium as well. The other categories are not Nash equilibria: in both a tragic ending and a missed opportunity profile, only one player would need to change their strategy in order to achieve a payoff of (+10,+10).

The butt-head profile is not subgame perfect: by definition, it implies that if one player gave in, the other player would stay firm and receive a payoff of +5 instead of giving in and receiving a payoff of +10. Therefore, the subgame perfect strategy profiles are all happy endings. (It's easy to construct a happy ending profile that isn't subgame perfect, but it's also easy to construct one that is.)

Neither tragic ending nor missed opportunity profiles are even rationalizable. In a tragic ending, one player plays a strictly dominated strategy. In a missed opportunity, for both players' strategies to be rational, each has to assume that the other one would play a strictly dominated strategy. Thus, the rationalizable strategy profiles are exactly the Nash equilibria.

In summary:

Beyond rationalizability
The interesting point is now that the canonical missed opportunity profile, which seems like it may be played regularly in real life, does not even belong to the class of rationalizable profiles, which seems to be the widest solution concept (class of strategy profiles) ordinarily studied in game theory.

Rationalizability can be informally described as the assumption that all players are rational, and that this rationality is common knowledge (player A knows that player B is rational, player B knows that player A knows that player B is rational, and so on). As mentioned above, though, in the canonical missed opportunity profile, both players play the unique Minimax strategy available to them, that is, player A's strategy is the best response under the assumption that player B will (irrationally) be trying to maximize player A's loss, rather than maximize player B's win (and vice versa).

More precisely, each player's strategy is a best response in every subgame, not only in the game as a whole. This distinguishes the strategy from the "never give in" strategy, which is a best response in the overall game, but it is strictly dominated in the subgames where the other player has given in first. We can therefore characterize the canonical missed opportunity profile, along with some happy ending profiles, by the property that no player plays a strictly dominated strategy in any subgame.

To do
We could describe the real-world situations alluded to in terms of a Bayesian game in which each player can be of a type for which hurting the other player would have a highest possible payoff. Analyze this description in terms of game theory; think informally about whether it seems to describe the real-world situation better or not.

To maybe do
Informally, it seems like playing the Minimax strategy might be the best thing to do if you have no information at all about what strategy the other player might follow. Can this be formalized? Is the formalized version true?