Real-life strategies
Theory of Mind
How humans may react can also be predicted using the paper by Verbrugge and Mol [8]. This paper suggests that people may not use high order Theory of Mind: they do not reason to the degree that their number of choice approaches 2. Or, they may assume that the other player does not use high order Theory of Mind, so the other player will not reason to that same degree. Of course it is also possible that both cases are true.
The paper also suggests that how pragmatic the player will be is dependent on how cooperative the player is. If the player wants to maximize the common profit, it will naturally choose a high number (probably 100). But if the player is uncooperative, for example if he or she has a strong desire to 'win' the honesty reward, then he or she will probably show less pragmatic behaviour, assuming having a higher claim is less useful than being the one that obtains the honesty reward.
Another situation that may occur is that one or both of the players will assume cooperative or even altruistic behaviour (to some degree) from the other player. This makes it more attractive to choose a higher number, even if the player has a desire to obtain the honesty reward.
Game Theory Experts
Another approach to see what the logical result of the Traveler's Dilemma would be, when played by human players, is given by Becker et al. [9]They wanted the participants that play the game to make the most logical choice. Therefore, they addressed members of the illustre Game Theory Society. These would, of all people, make the most logical decision. To encourage this, one of the participants would receive a prize, equivalent with the performance of their strategy. And another, for the goodness of fit of their belief of the other participants' strategies.
Participation to the experiment consisted of two steps. First of all, all participants had to specify what their belief was about what the other participants were going to enter. This was done by giving a probability value to each value in the strategy space
. Second, all participants had to specify their own strategy.
The participants had the choice to do two strategies: they could either play on a single value (a pure strategy), or assign certain probabilities to a number of values (a mixed strategy).
Results
51 participants entered a strategy, from whom 47 entered a belief of the average strategy.The average choice of the participants in Becker et al.'s experiment is shown in Figure 3.1. The large averages at 31, 49 and 70 come from pure strategies with those values. As can be seen, some of the participants actually played the Nash equilibrium of 2. Also, 10 participants entered the 'irrational' strategy 100. The authors call this strategy irrational, because the strategy 100 is strictly dominated.
The Nash equilibrium was the least profitable strategy, as can be seen in Figure 3.2. Note here that the payoffs shown are directly dependant on the strategies of the other players. Therefore, the ultimate strategy differs per experiment.
![\includegraphics[width=\textwidth]{strategies.eps}](img99.gif) |
![\includegraphics[width=\textwidth]{payoffs.eps}](img100.gif) |
A nice result in the beliefs given by the participants, was that only 17 (out of the 47) of the participants played a strategy that coincided with their belief of the average strategy. From this we can deduce that some of the participants deliberately avoided playing the average strategy, to gain a higher payoff.
Now, Becker et al. claim that the Traveler's Dilemma becomes a game of incomplete information. This must be the case, because there is no single winning strategy. The only logical strategy that can be inferred is the Nash equilibrium, and they claimed that this is an irrational choice.
Next, Becker et al. disregard all strategies but those that fall (partially) into the interval ![$[94, ..., 99]$](img101.gif){kind=small}
. Disregarding the players that chose the strategy 2 or the strategy 100, of the remaining 38 participants, 27 (more or less) adhere to this interval model.
After neglecting all strategies outside this interval model, a weight of 0.281 is not accounted for, the rest is contained in the model. They now think that the resulting model is a reasonably good approximation to the weight of each of these points becoming an equilibrium.
Conclusions
Becker et al. conclude that their resulting model is a good one for individual predictions, because of the fact that all participants were experts in game theory. They therefore conclude that the fact that their interval model is reasonably high up the strategy space, indicates that these high values are not because the participants did not understand the game (and therefore did not end up with the Nash equilibrium 2), but the high result is the result of a robust pattern of behaviour.While analyzing the game as a game of incomplete information, the authors still have no arguments for the fact that almost 20 percent of the participants chose the, to them, irrational strategy 100. However, they note that, when no player would play the irrational strategy 100, all players would end up in the Nash equilibrium of 2.
We can reason that, when there are participants that play the strategy 100, there is no need for the other participants to go much lower than the honesty reward below that. As shown in the second chapter, the choice for 100 is strictly dominated, so there is no reason to take it. But if it is taken (for example as a cooperative strategy) then it would be foolish to go much lower, and get a lower payoff than you could have had.
Whilst if the strategy 100 was abandoned, the race for the just-a-bit-lower claim would start, and finally, all players would indeed end up in the Nash equilibrium of 2. But this reasoning is not mentioned by the authors.
We think that the way that Becker et al. handled the problem is curious. The way they analyze the data is good, but how they handle the observed data, and how they think about the contestants is remarkable.
First of all, they say some of the result is irrational, while they tried their best to let the participants be as rational as possible. Next, they discard part of the results, taking a range that fits them, and then saying that the results can be fitted well inside that range. While a more statistical approach would be to first try to analyze what the probable result would be, and then actually compare the results with the model created at first.