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5

So $h$ is just some history of the game. Consider the following game, where Player 1 first decides Heads or Tails, then depending on his choice, a coin is flipped whose outcome and probabilities depends on the choice of Player 1. An example of a history $h$ is $h = Heads$, i.e. after heads is chosen. Consider the very beginning of the game, before Player 1 ...

3

There is no guarantee that a random game can be solved by an easy trick. (For an example see Kuhn poker.) Did you study something like this in class? If yes, was there a step-by-step method the teacher followed? Perhaps the steps were not designated as steps, but if you look back on it, you may realize that is what they were? If yes, you should probably ...

2

A direct revelation mechanism is one in which a player's type space is also their action space ($X_i=T_i$ for all $i$) and the outcome function is the same as the social choice function ($a(t)=f(t)$ for all $t\in T_1\times\cdots\times T_n$).

3

Let $n \geq 2$. Observe that $$\mathbb{E}[u_i] = \mathbb{E}[u_i|v_i > r]P(v_i > r) + \mathbb{E}[u_i|v_i \leq r]P(v_i \leq r) = \mathbb{E}[u_i|v_i > r]P(v_i > r)$$ since $\mathbb{E}[u_i|v_i \leq r] = 0$. Moreover, $P(v_i > r) = 1 - r$ if the values are standard uniform. So to compute the expected payoff $\mathbb{E}[u_i]$, all that remains is ...

2

Consider bidder $i$. Let $p = \max_{(2)}\{b_{-i}\}$. By bidding $v_i$, she wins if $v_i > p$ and not if $v_i < p$ (and indifferent if $x_i = p$). Suppose, however, she bids $z_i < v_i$. (i) If $v_i > z_i > p$, she still wins the auction and still gets $v_i - p$. (ii) If $p > v_i > z_i$, she still loses the auction and still gets $0$. (...

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