“Clocks and Trees: Isomorphic Dutch Auctions and Centipede Games,” J. C. Cox & D. James (2012)

There has been a substantial amount of effort by experimental economists showing where lab subjects take actions other than Nash equilibria play in common games. Folks appear to give away money in ultimatum games, bid too high in Dutch auctions, wait too long to end the centipede game, and so on. The most common response to these experimental facts is to propose alternative utility functions (such as forms of social preferences) or alternative equilibrium definitions that incorporate particular types of errors (like the Quantal Response).

Cox and James dig back into Selten’s famous Chain Store Paradox paper for an alternative explanation. (An aside: I imagined this had to be the most cited paper ever to appear in Theory and Decision; it is.) Selten points out that the very first thing an agent in a game must do is understand what her options are, what information can be transmitted, and what the consequences of any tuple of actions will be. Such information must generally be deduced by an agent from the description of the game; it is not obvious. When we teach game theoretic reasoning, we generally make such complicated deduction easier through the use of an extensive-form game tree. That is, how well students will do on a game theory exam is absolutely affected by how the information in an identical game is presented.

Cox and James discuss this point in the context of Dutch auctions and centipede games. In a Dutch auction, agents have a private valuation for a good, the auction starts at some amount higher than that valuation, and then the price drops until some agents “bids”, winning the good at that price, and getting payoff equal to her valuation minus the bid price. Alternatively, we could write a (discrete-interval) Dutch auction out as a game tree. If you have valuation 6 and you bid when the auction is at “5”, you win a dollar. Every possible bid by each player can be written out in the tree, along with consequences. It turns out that presenting information in such a tree led to actions which are roughly the same as predicted by Nash equilibrium, whereas the traditional declining-clock Dutch auction without a tree led to the overbidding seen in other experiments.

The centipede game is a bit different. Traditional experiments present the game as an extensive-form game tree. But there is a nice way to run a centipede experiment using a clock. Start at state 0, giving player 1 the option to stop the game and win his valuation. If the player does not stop the game within 10 seconds, shift to state 2, where player 2 has the option to stop the game and win her valuation plus 2, giving player 1 nothing. If she does not stop the game after 10 seconds, go to stage 3, and so on. The clock version of the centipede game led almost all subjects to stop at 0, which is the unique Nash equilibrium.

There are two things you might take out of these results. One is that play in games involves both the player’s strategic reasoning as well as their comprehension about how the game operates. Both are important. It makes me skeptical of some experimental results showing non-equilibrium outcomes. Second, you might imagine that we could incorporate learning about the options available to players as they play repeatedly into an equilibrium definition. The draft of this paper I read did not refer to the 1990s learning-in-games theoretical literature, which absolutely has discussed this point, but perhaps the final Econometrica version does.

Working paper (IDEAS version). Final paper published in March 2012 Econometrica.