Una visión puntual: los cañones de guerra

The most common type of auction found online is a second-price auction. A second- price auction awards the object of the auction to the highest bidder, with the price being determined by the second-highest bid. An auction with n participants can be described as a game among n players. Each player might value the object of the

auction differently. Let virepresent the monetary value of the auction object to player i. Each participant’s behavior must depend on the payoffs that he or she will receive for each possible strategy. More formally, a game is a collection of available actions for all players and a set of payoffs, with one payoff value corresponding to each possible collection of actions by all players. Thus to specify a game, we must list the players, the actions available to each player, and the payoffs for every result. First, let’s examine the case of a second-price sealed-bid auction, often referred to as a Vickrey auction. A sealed-bid auction means that no bidder is aware of other bidders’ bids at the time they place their own bid. Because players cannot observe the bids of others, a player cannot condition his actions on the actions of other players. In this case, bidder i’s payoffs can be characterized as

πi=

vi− p if xi> p 0 if xi≤ p,

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where p is the second-highest bid and thus the realized price for the object, and xiis bidder i’s bid. First, consider a bid with xi< vi. In this case, if the bidder wins the auction, xi> p, then the bidder obtains the value from the good but loses the auction price, resulting in a net gain of vi− p > 0. Because this value is positive, the bidder could only benefit from wining the auction in this case.

Alternatively, suppose the bidder lost the auction, xi< p. In this case, the bidder obtains zero benefit. If the price is above his bid, but below his valuation, v_i> p > x_i, then increasing his bid to any value above p will result in winning the item and obtaining vi− p > 0, which is strictly better than if he had bid below p. Thus, the bidder might fail to purchase the item when it is in his best interest to obtain it if he bid below his valuation and the realized price is also below his valuation of the object.

Suppose instead that the bidder bid above his value, xi> vi. If his bid is above the price, then he wins the item, obtaining a value of vi− p. This is positive if vi> p, but nonpositive otherwise. Thus, if the price realized is over the bidder’s value but below his bid, xi> p > vi, the bidder is strictly worse off than if he had bid below p. Thus, if the bidder bids above his valuation of the object, any realized price above his valuation could result in his obtaining the item at more than it is worth, resulting in a loss.

Finally, suppose the bidder bids exactly his valuation, xi= vi. In this case, if the realized price is below his bid, he will obtain vi− p > 0, which is the most he could receive under any possible bid. Alternatively, if the realized price is above his bid, then he obtains 0. Increasing his bid could obtain the item, but it would necessarily result in a loss because he would pay more than it was worth to him. Thus, in this case, 0 is the best outcome that could be obtained under any bid. Bidding one’s value results in obtaining the highest payout of any strategy, no matter what price results from the combination of all participants’ bids. When a strategy produces the highest payout to the individual of any possible strategy under all possible outcomes, the strategy is called a dominant strategy. In this case, each player in the Vickrey auction has a dominant strategy to bid his or her valuation of the object no matter what anyone else does. Thus a rational model would predict that people bid their valuation no matter what. In this scenario, the bidder

who values the good the most wins the auction and pays the second-highest valuation for the item.

Suppose instead that people could observe others’ bids and could sequentially increase their bid if they decided. In this case, whenever all other bids were below the bidder’s valuation, the bidder should bid above all others. If he subsequently saw further bids from others above his previous bid but below his valuation, he should increase his bid again. However, if any bid occurs above his value, he should cease to bid. This will lead to an outcome where the bidder who values the good the most will eventually outbid all others and will pay the second-highest bid, which will be given by the second-highest valuation, just as in the sealed-bid auction. Thus, there is no advantage to gaming a second-price auction. One could just as easily bid one’s valuation of the item at the onset of the auction and wait to see if others valued the item more or less.

Finally, consider afirst-price auction. A first-price auction is an auction mechanism where the top bidder wins the item but must pay his own bid for the item. In this case, the winner’s payout can be described as

πi=

v − xi if xi= p 0 if xi< p,

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so that the winner obtains the value of the item, minus his bid if his bid is the realized price, and obtains zero if his bid is less than the realized price. If this were afirst-price open-bid auction, then the bidder could observe the bids of others. If there were any bids above his valuation, he would choose to bid at a lower value, lose the item, and gain zero. Alternatively, if the highest bid by any other player were below his valu- ation, he would choose to bid just slightly higher than the next highest bid, so as to obtain the item, gaining the difference between his valuation and his bid—essentially the next-highest bid. Thus, thefirst-price open-bid auction results in outcomes that are much like the second-price auction. Everyone should bid up to their assessed value except the bidder with this highest valuation, who bids as high as the second-highest value and obtains the item. Thus, in a wide range of auction types, the bidder valuing the object the most should win and should pay an amount equal to the second-highest valuation.

In more-general sets of games it is not always possible tofind a dominant strategy. The primary tool to make predictions when the player’s outcome depends on the actions of others is the concept of the Nash equilibrium. To define the Nash equilibrium, we needfirst to define a few useful terms.

Define a node as any point in the game where a decision can be made by player i and the information available to the player at that point. For example, in playing a game of tic-tac-toe, the beginning of the game places player 1 at a node where she can choose to place her mark in any of nine squares, with no knowledge of where player 2 will choose to respond. If player 1 places her mark in the center square, this places player 2 at a node where player 1 has placed a mark in the center square; he now knows where player 1 placed her mark and can choose to place his mark on any of the remaining eight squares. Had player 1 placed her mark in the upper left square, we would have arrived at a

different node where eight squares were available, but now the upper left square is taken, and the center square is available.

A strategy is a collection of the decisions a player intends to make at each possible node of the game should that node be reached. It is important to differentiate between a strategy and an action. One action might be“place an X on the center square.” A strategy would be a long list of actions, one for each node, such as“if player 1’s first move is to place an O in the top left corner, then myfirst move will be to place an X in the center square; if player 1’s first move is to place an O in the upper middle square, then my first move . . . ,” and so on, describing every action in response to every action or history of actions in the game. Letπ_i siS− i be the payoff received by player i for playing strategy si, when all other players are playing strategies represented by the symbol S− i.

The Nash equilibrium is a collection of strategies S= s1, , sn , such that for each player i, πi siS− i ≥ πi si S− i , where S= si S− i. Intuitively, the Nash equilibrium is a set of strategies, one for each player, such that each player is maximizing his or her payoff given the strategies of all others involved. Thus, given the strategies of all others, any single player should not be better off for choosing a different strategy. In the second- price auction, a Nash equilibrium is given by each bidder bidding his or her valuation. In the open-bidfirst-price auction, a Nash equilibrium is given by each bidder choosing to bid his own valuation unless he is the top bidder, in which case he bids just above the second-highest bidder. In general, if any player has a dominant strategy, all Nash equilibriums must require this player to play his dominant strategy. The Nash equilib- rium is widely considered the basic requirement for rationality in a game.

EXAMPLE 5.1 Second-Price Auctions

John Kagel, Ronald Harstad, and Dan Levin conducted a set of experiments to test the predicted Nash equilibrium under several types of auctions. Among these, they tested the Vickrey auction mechanism. Participants were randomly assigned a private valuation for winning the auction—in other words, each was assigned a value they would receive if they won the auction and each participant’s value was different. This valuation took the form of an amount of money they would receive if they happened to win the auction. Participants were informed of the auction mechanism and they engaged in several rounds of Vickrey auctions. In each round, the participant was assigned a new private valuation for winning. The Nash equilibrium strategy is to bid one’s private value for the auction, but the bids were, on average, substantially above the private valuation assigned. On average, the bids were between 12 percent and 20 percent above the private value, with larger percentages when the private value was lower. Participants experienced dozens of Vickrey auctions, with no evidence that their bids were con- verging to the dominant strategy of bidding one’s valuation. This led to the curious condition that in 80 percent of the auctions, the price (determined by the second-highest bidder) was above the second-highest valuation in the auction, with 36 percent of the auction winners paying a price that was above their own private valuation. These 36 percent were winners only in the nominal sense of the word, as they were the only participants in these auctions to lose money.