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Playing 2 is the weakly dominant strategy for both, no?

I hesitate to try a table in the comments, but here goes. The payoff for each player. The opponent plays (0, 1, 2).

Player's play\Payoff

0 ( 0, 0, -2)
1 (-1, 0, -1)
2 (-2, 0, 0)

Note that this game is an example of Game B, with the payoffs doubled. :)

Nick,

Usually, we use equilibrium to indicate what the model predicts will happen, but it's important to note that this is not the definition of equilibrium. In some cases, the distinction is not that important but in others it is. The equilibrium in the prisoner's dilemma is highly predictive. In Min's game above, the various equilibria tell us little about what will actually happen. There are other hairs to split here too. In a long-run growth model, technically the model doesn't predict that the steady-state will ever "happen" exactly (depending on the details of the model) unless you happen to start there. Yet we still basically treat it as what the model predicts because it predicts that we would *approach* it.

Regarding your question about preventing jumps from one equilibrium to another, I don't think there is any general answer. There is nothing inherent in the definition of NE that deals with jumping from one to another. Like I said, the definition just amounts to a state in which nothing is expected to push it away. If you are talking about what "pushes" something to one equilibrium or another, you are going beyond the concept of equilibrium. This is why it's important to keep the proper definition of equilibrium in mind. BTW, there is nothing wrong with talking about what pushes something to one equilibrium or another. The competitive market equilibrium is just a NE to a simultaneous-move game but it makes for a compelling *predictive* model in part because we can tell a story about what would happen if the price were higher or lower than the equilibrium price and that story "pushes" the price toward the equilibrium. If, on the other hand, our story seemed to push the price away from that equilibrium, it would be a much less convincing prediction but it would still be an equilibrium.

Mike: " In a long-run growth model, technically the model doesn't predict that the steady-state will ever "happen" exactly (depending on the details of the model) unless you happen to start there."

Agreed. In a long run growth model, for example, there is a ("short run") equilibrium path predicted by the model, and there may (or may not) be a "long run" steady state equilibrium, independent of initial conditions, that may (or may not) be stable.

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