In the beginning of the mahjong anime/manga Akagi, the titular character has just come in from what is called a game of chicken, which is described as follows:
Two cars head for a cliff at full speed. It's a game of reckless disregard for life where the person who first brakes loses.
No conditions for ties were given in the anime, but I will assume that:
Driving over the cliff is the least preferable outcome regardless of what happens to the other person
Losing is preferable to driving over the cliff
Being in a tie is preferred over losing, and winning is the most preferred outcome
I believe that the games aren't equivalent:
Unlike the description of the game of chicken given on Wikipedia, there is a weakly dominant strategy in the game in Akagi (i.e. braking at the edge of the cliff). But I'm not sure if the existence of weakly dominant strategies are preserved under isomorphisms of games.
The Akagi version of chicken has a pure Nash equilibrium where both players take the same action (and both obtain an outcome that isn't their most preferred outcome). This outcome (in terms of preference rankings) doesn't exist in the standard formulation of the game of chicken. However, thinking about it, it might be better to view the Akagi case as something continuous that might then permit mixed strategies (with regards to when the player chooses to brake).
Is my reasoning here correct, or is there a better way to view this?
Update: I should add that in the Akagi version, the choice is when to brake, which lies on a continuum in some interval of $\mathbb{R}$. (I'm not entirely sure how to formulate this mathematically, since we also need to allow for the possibility that the player drives off the cliff, but this is beside the point.) However, as previously stated, braking right at the edge should be weakly dominant, at least from an intuitive perspective.
Conversely, in the standard version, there is a binary choice ("swerve" or "straight") at a single point in time. Unless it's possible to rule out all but a few cases in the Akagi version, it would seem that the games can't possibly be equivalent, since we can't "reduce" the Akagi case to the normal game of chicken. (There should be no isomorphism between them if the cardinalities of the set of choices are different.) But that's exactly the thing that I'm not completely sure about.
Moreover, I'm not sure if I'm analysing the equilibria of the Akagi case properly, particularly since it strikes me as a scenario where mixing might do better (but I'm not familiar with mixed strategies in continuous cases, and I'm too busy at the moment to test somewhat "simpler" cases).
