Interpretation of the Myerson (1981) Bidder Utility (Optimal Auctions, Mechanism Design)

Question

In Myerson 1981, the bidders' utility function does not make sense. $p_{i}$ is a probability of winning the auction, $t_i$ is the bidder's type (or valuation of the item), and $x_i$ is the payment. $r$ is your bid or reported type. In the paper, the utility function of bidder $i$ is: $$u_i=p_i(r) t_i-x_i(r)$$ This is very strange. The payment term $x_i$ is (strangely) not being interacted with the probability of winning $p_i$. This makes it appear that $i$ pays $x_i$ unconditionally, (that is, even if $i$ loses).

PART I: What am I missing? Shouldn't it be $u_i=p_i(r)(t_i-x_i(r))$? I realize my formulation may be harder to solve, but I think it's the more typical auction behavior: pay only if you win.

PART II: I asked a colleague about the above. He is knowledgeable but not authoritative in this area. The colleague claims my interpretation is correct ($x_i$ is paid unconditionally in the formulation above).

However, he also says it is possible to recover the conditional payment scheme once you've solved for the unconditional payment scheme. To solve it, let $c_i(r)$ represent the payment conditional on winning. The bidder's utility is $u_i=p_i(r)(t_i-c_i(r))$. For the bidder to have the same expected utility under the conditional and unconditional payment schemes, then $$p_i(r) t_i-x_i(r)=p_i(r)(t_i-c_i(r))$$ $$\implies c_i(r)=\frac{x_i(r)}{p_i(r)}$$ So the conditional payment is the unconditional payment, scaled by the probability of winning. This seems intuitive to me, but hey: My colleague is knowledgable but not authoritative. Maybe there's something we're missing. I wish this were made explicit in a paper somewhere.

Answer 1

First, the formula you write does not occur in the paper. Expected payoffs are given in 3.1, the notation there is different. Second, Myerson writes on page 61 that "$x_i(t)$ is the expected amount of money which bidder $i$ must pay to the seller." So this is the expected payoff (conditional on the type) and already incorporates the probabilities.

The preferences do not specify how the actual mechanism relates to payments and winning probabilities. Lemma 2 proven in this paper can be used to show that (under some standard symmetry assumptions and in the unique monotone equilibrium) the expected payoff of a bidder in a first price auction (where only the winner pays their bid) and in an all-pay auction (where everyone pays their bid) is the same. In the first case, the payment is conditional, in the second it is not. Both are allowed here.