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4

There is another way to compute the symmetric BNE in increasing strategy. Let $U(v)$ denote the expected utility of a player in equilibrium when his type is $v$: Given that the bidding strategy is increasing, a player with type $0$ will get the good with probability zero. Thus he/she must bid zero and $U(0) = 0$. For any other $v > 0$, the probability ...


3

Almost all textbooks on game theory include a part on cooperative game theory and therefore also treat Nash bargaining. A random sample ($n=3$) from my shelf produced this one, this one, and this one.


3

If you have in mind infinite populations I'd suggest Carlos Alós-Ferrer (1999) Dynamical Systems with a Continuum of Randomly Matched Agents, Journal of Economic Theory 86 (2), 245-267.


7

Let $N=2$ and for $(x,y)$ and $(p,q)$ in $[0,1]^2$ let $d_{p,q}(x,y)$ be the Euclidean distance between $(p,q)$ and $(x,y)$, i.e. $d_{p,q}(x,y)=[(p-x)^2+(q-y)^2]^{1/2}$. Choose $k>0$ such that $k<a-b$ and $k<b$. Now define $$u(p,q):=\max\{ -d_{p,q}(a,a),-d_{p,q}(b,b),k-d_{p,q}(b,0),k-d_{p,q}(0,b) \}.$$ This function is continuous as a maximum of ...


2

We cannot judge if your answer is correct because we don't see the game tree. First, I would not say that "we can rule out this equlibrium as a possible pooling PBE by the intuitive criterion" because the intuitive criterion is simply a refinement. The PBE is still an equilibrium - it's just that we can say that it appears "unreasonable" ...


2

Think about the incentives of player $i$: If he knew that no one else helps, he'd want to help. If he knew that at least one other player helps, he'd rather not help. A single player helping would make everyone happy, but no one wants to be that single player, because it's costly to help. This is the classical problem of finding a volunteer, so that's why ...


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