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  • Construct a pair of startegies for the ultimatum game ($T=1$ bargaining game), that constitutes a Nash Equilibrium and together support the outcome that there is no agreement reached by the two players and the payoffs are zero to each. Show that the disagreement outcome can be supported by a Nash Equilibrium regardless of the number of bargaining periods.

I understand that in a bargaining game if $T=1$ then any division of surplus $x^*$ belonging to $[0,1]$, $(v_1, v_2)=(x^*,1-x^*)$ can be supported as a Nash Equilibrium.

In the question given above, I'm supposed to find a strategy such that there is no agreement reached and both players end up getting $0$ each. I have some trouble thinking about such strategies and whether it would constitute a Nash Equilibrium. One pair I could think of is: Player $1$'s strategy is to offer, say, $(x^*,1-x^*)=(0.6,0.4)$ and Player $2$'s strategy is Accept any offer $(1-x^*)$, $>$ or $=$ $0.5$. Hence we see that Player $2$ will reject the strategy. However, I cannot understand how a Nash Equilibrium startegy can be constructed (since the above stated startegy is clearly not optimal).

Another strategy could be, Player $1$ proposes $(0,1)$, and Player 2's strategy is Accept only strictly positive offers, i.e., reject $0$. However, this strategy cannot be Nash Equilibrium because the first mover has no Best Response.

Please help me understand the question. Hints for the next part would also be appreciated. Thank You!

Reference: Game Theory: An Introduction, Steven Tadelis

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Consider:

  • Proposer offers $0$
  • Receiver always rejects the offer regardless of the amount

You should be able to argue that this is a pair of mutually best responding strategies for $T=1$. The $T>1$ cases follow a similar logic.

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