# Rationality and the winner's curse

Classical economic theory generally considers behavior as rational, and processes empirical observations into theoretical frameworks under the infamous rationality assumptions¹, which essentially boil down to the imposition of a "revealed preferences" constraint on the actions of the agent at stake. In layman terms, this means that whenever one acts, one does so in an optimal fashion, in one's own eyes, thereby revealing what his or her preferences are. For instance, if one orders a cup of tea instead of the coffee alternative, we may learn that he or she favors the former over the latter, assuming no budget concern from either choice. One may extrapolate this characterization to more concrete scenarios: for instance, a CEO is expected to manage his or her company in an optimal fashion, which is equivalent to stating that firms maximize profit given their knowledge of competition, costs, and profitability prospects. A practical example is the following: governments often publish auctions, say, for the realization of real estate projects. Any number of competitors may step in and compete for the rights to carry out the construction; common sense informs us that the auction's participants have a good reason to be such, namely, cash expectations. Surely the winner must earn profit… well, it turns out not always to be the case. We will come back to this later.

The revealed preferences axiom has its limits. To realize this, it suffices to examine a most mundane situation: grocery shopping. Thousands of goods lie on most supermarkets' shelves. To select the best cart, one should optimize the division of the weekly budget given the available products, their prices, and of course personal requirements. Assuming 1000 different products and a binary "buy/do not buy" choice for each one of them would imply the comparison of $2^{1000}$ baskets, which is lies far beyond the computing power of a human brain (and of anything else, for that matter). This suggests the use of heuristic rules in decision making, such as the following:

• One knows whether lavender-scented parquet cleaners are needed or not, at any given time; whenever it is not the case, they are not considered at all.
• One does not select the best achievable basket, but a reasonably good one, according to some threshold. By this, I mean that, denoting a basket $b$ and a utility function $u(b)$, one does not select $\arg\max_{b} u(b)$ but rather some $\tilde{b}\in\left\{ b|u\left(b\right)>\underline{u}\right\}$, with $\underline{u}$ said minimal threshold.

A whole branch of economic research deals with what is called, in academic jargon, "bounded rationality," which expands on such concepts². In practice, optimization problems should arise whenever information is not complete, that is, when the agents make decisions under uncertainty, since humans generally fail at accounting for probabilistic world states. Empirically, to our purposes, an interesting phenomenon stems from such calculation errors: the winner's curse, which roughly states that in auctions, winners exhibit a tendency to bet more than what winning is worth to him or her.

Going back to the construction rights auction helps providing some intuition for the result. Assume that the government sells the construction rights to the highest bidder at the proposed price (i.e., conducts a "first-price auction") and that, for the sake of simplicity, construction firms usually know how to correctly assess the expected revenue from a project, as well as the costs it implies, due to their wide knowledge of the market. Along those lines, profitability should display low variation across the players: the industry being complex, all competitors would be experts, yielding aggressive competition. All bids are submitted "sealed:" they are private information to each one of the bidders. Under our settings, the average bid should lie near the actual profitability of the project. It is straightforward to conclude that the winner is likely to be at a loss: the average being close the the true value implies that the winner overbids.

We finally present a simple game theoretic example (which may safely be skipped).
Let a Bayesian game $G$ have common value payoffs $v\left(t_{1},t_{2}\right)=\beta_{1}t_{1}+\beta_{2}t_{2}$ with $t_{i}$ the type of firm $i$, $i\in{1,2}$ (which easily generalizes), $\beta_{i}>0$. The type denotes the perceived profitability, is uniformly distributed in $[0,1]^{2}$, and $v$ simply denotes some weighted average of these subjective appraisals.
Suppose $\beta_{1}=\beta_{2}=100$ and consider the strategy $s_{i}\left(t_{i}\right)=100t_{i}$. If $t_{i}=0.1$, $s_{i}\left(t_{i}\right)=10$, and $\mathbb{E}[v|t_{1}=0.1]=100\cdot0.1+100\cdot\mathbb{E}[t_{2}]=60$. An error that firm 1 might make would be to compute the expected revenue conditional on its own type, omitting the critical conditioning on winning (that is, on offering the highest price), in which case, given that $\mathbb{E}[v|t_{1}]=60$ and that $s_{2}\left(t_{2}\right)=100\cdot t_{2}$ (denoting $a_{1}$ its strategy): $\arg\max_{a_{1}}\Pr\left(a_{1}>s_{2}\left(t_{2}\right)\right)\cdot\left(60-a_{1}\right)$ $= 30$.  Yet, it would be misguided for firm 1 to bid $s_{1}=30$$100\cdot 0.1+100\cdot\mathbb{E}[t_{2}|t_{2}<0.3]=25<30$, thus losing 5.

The central tenet here is the common value: if there is large variance in the assessments the firms make, there is no reason the highest bid would not be worthwhile.

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¹ While this is absolutely contingent on our discussion, it constitutes a good opportunity to sneak in the Mas-Colell, Winston & Green (1995), section 1.B reference for curious non-academics who which to enhance their orthodoxy-bashing apparatus. The exposition provided thereby is clear and critical enough to reference Kahneman & Tversky (1984) outright.

² A good introduction to this subject is Kahneman's Nobel lecture (2002). A more in-depth reference is Rubinstein's Modeling bounded rationality (1999).

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