Some years ago, my son introduced me to the "Game of 21". The first player chooses a number (1 or 2), and then players take turns incrementing the count by 1 or 2. So, for example, if the first player chooses 1, then the second player could choose either 2 or 3, but if the first player chooses 2, then the second player could choose either 3 or 4. The winner is the player that chooses 21. My son beat me handily, but he knew the winning strategy, which is to always choose a multiple of 3, if one is available. To see why that's a winning strategy, we can work backwards from 21. If you choose 18, then your opponent must choose either 19 or 20, in which case you can choose 21. If you choose 15, then your opponent must choose either 16 or 17, in which case you can choose 18, and then 21 after their next choice. And so on (15, 12, 9, 6, and 3). It turns out that there is a clear second-mover advantage in the Game of 21, since the second player can always choose 3, regardless of what the first player chooses, and the first player can never choose a multiple of 3 if the second player does so.
The winning strategy seems obvious when it is explained to you, but it is far from obvious to most people before the game begins. How long does it take for people to figure it out, and would a shorter game (say, a "Game of 6") help? Those are the research questions that this 2010 article by Martin Dufwenberg (University of Arizona), Ramya Sundaram (George Washington University), David Butler (University of Western Australia), published in the Journal of Economic Behavior and Organization (ungated version here), tackle. Using a sample of 72 research participants, they had 42 of them pair up (in a round-robin format) for five rounds of the Game of 21 (G21) followed by five rounds of the Game of 6 (G6), and 30 of them did the reverse (five rounds of G6, then five rounds of G21). Essentially, they test whether playing the simpler G6, where recognising the multiple-of-3 winning strategy is much easier, helps players to recognise the winning strategy for G21. Indeed, that's what they find. Looking only at players who play second (the 'Green' player in their wording), since only those players have a dominant strategy, they look at the proportion of players playing a 'perfect' game.
First, Dufwenberg et al. note that:
...most subjects playing five rounds of G6 realize that G6 may be solvable by rational calculation...
...most subjects playing the Green position in G21 for the first time do not immediately figure out that choosing multiples-of-three is the best they can do... Across treatments, in G21, only 49 of 179 games (27%) are played perfectly... The rates of perfect play are especially low in the early rounds of the G21-then-G6 treatment (e.g. 2 out of 20, or 10%, in round 1).
Then, turning to their main research question, Dufwenberg et al. find that:
Green players play G21 perfectly in the G6-then-G21 treatment 37% of the time, compared to 21 percent in the G21-then-G6 treatment. This difference is significant at the 5% level (Z statistic = 2.20).
They also show that, among players who appear to have figured out the winning strategy in G21, players who played G6 before G21 choose the winning strategy earlier, on average, than those who played G21 before G6. That suggests that we can learn how to optimise in difficult strategic situations if we are first presented with similar but simpler situations.
An interesting side-point of the Dufwenberg et al. paper was the reference to level-k reasoning. Level-k reasoning refers to the number of steps of reasoning a decision-maker is capable of undertaking. As they note:
...level-0 players may choose randomly across all strategies. Level-1 players assume everyone else is level-0, and best respond; level-2 players assume everyone else is a level-1 player, and best respond; etc...
G6 and G21 don't really require much in the way of steps of reasoning, because once you realise what the winning strategy is, it doesn't matter much what the other player does (unless they don't know the winning strategy).
However, one game that does test level-k reasoning is the 'beauty contest'. Each player must choose a number between 0 and 100, and the winner is the player who chooses the number that is closest to two-thirds (or some other fraction) of the average of all guesses. I played this game many times with students when I was teaching a third-year Managerial Economics and Strategy paper. Level-0 reasoning would lead to a player choosing randomly. If all players did that, then the rational choice for a Level-1 reasoning player would be to choose 33 (two-thirds of the average of 50). However, if you believed that everyone else was a Level-1 reasoning player, making you a Level-2 reasoning player, then you should choose 22 (two-thirds of 33). And, if you believed that everyone else was a Level-2 reasoning player, making you a Level-3 reasoning player, they you should choose 14 (two-thirds of 22). And so on. The Nash equilibrium here is for everyone to choose 1 (or 0, depending on how the game is scored). However, the outcome is never that the winning score is 0. From memory, the winning score in my class was always around 10-20.
How many steps of reasoning do people engage in? That question has drawn a lot of research attention. One interesting aspect is how we early in life we develop level-k reasoning. That's the topic of this new article by Isabelle Brocas and Juan Carrillo (both University of Southern California), published in the Journal of Political Economy (ungated earlier version here). Brocas and Carillo created a very simple three-player game that could be easily solved by backward induction (for those who understand some game theory). As they describe it:
...subjects were matched in groups of three and assigned a role as player 1, player 2, or player 3, from now on referred to as role 1, role 2, and role 3. Each player in the group had three objects, and each object had three attributes: a shape (square, triangle, or circle), a color (red, blue, or yellow), and a letter (A, B, or C). Players had to simultaneously select one object. Role 1 would obtain points if the object he chose matched a given attribute of the object chosen by role 2. Similarly, role 2 would obtain points if the object he chose matched a given attribute of the object chosen by role 3. Finally, role 3 would obtain points if the object he chose matched a given attribute of an extra object.
The accompanying Figure 1 in the paper helps to understand the game (although the figure is in black-and-white, and the description refers to colours, which doesn't help as much as it could!):
Player 3 is asked to match the shape, so they should choose the dark square C. Player 2 is asked to match the colour that Player 3 will choose, so they should choose the dark triangle B. Notice that Player 2 needs to undertake two steps of reasoning, working out what Player 3 is doing in order to work out what they should do. Player 1 is asked to match the letter that Player 2 will choose, so they should choose the light circle B. Notice that Player 1 needs to undertake three steps of reasoning, because they must work out what Player 3 will do and then what player 2 will d, in order to work out what they should do.
Brocas and Carillo run their experiment with a number of samples of children and young adults, and each research participant played the game 18 times (six times in each of the three positions). They expect to find:
...four types of individuals: R (subjects who always play randomly), D0 (subjects who play at equilibrium only if they have a dominant strategy), D1 (subjects who play at equilibrium when they have a dominant strategy and can best respond to a D0 type), and D2 (subjects who can play as D0 and D1, as well as best respond to D1).
And in terms of behaviour:
The predicted behavior is simple. R plays the equilibrium strategy one-third of the time in all roles, D0 always plays the equilibrium strategy in role 3 and one-third of the time in roles 1 and 2, D1 always plays the equilibrium strategy in roles 2 and 3 and one-third of the time in role 1, and D2 always plays the equilibrium strategy.
Their first study involves students from third to eleventh grade from a private school in Los Angeles, along with undergraduate students from USC. Classifying the research participants into the four types outlined above, Brocas and Carillo find that:
Subjects either recognize only a dominant strategy or always play at equilibrium. Also, some very young players display an innate ability to play always at equilibrium while some young adults are unable to perform two steps of dominance.
In other words, there are no D1 players, as every player who can reason beyond one step can reason all the way through the steps. Then, looking only at the 234 grade school students in their sample, Brocas and Carillo find that:
Performance in roles 1 and 2 increases significantly up to a certain age (around 12 years old), and then stabilizes...
So, older students perform better, but only up to the age of 12 years. They then go on to replicate similar findings for a Los Angeles public school (where overall performance was lower) - there is no difference in performance from sixth to eighth grade (12-14 years old).
Finally, Brocas and Carillo study a sample of students from kindergarten to second grade. They simplify the game so that there are only two players (rather than three), and only two attributes (rather than three). With their sample of 117 children, they find that:
Equilibrium behavior is not significantly different between K and grade 1 in roles 2 and 3, and they are both lower than in grade 2 (p < .02, FDR adjusted).
The evolution of strategic behaviour as people age is interesting. That isn't quite what Brocas and Carillo are studying, since they don't follow the same children over time, but instead look across cohorts of different ages. However, it's hard to see how or why there would be a cohort effect here, so possibly they are observing an age effect. Interestingly, most of the improvement in strategic reasoning happens between the ages of 8 and 12 (second to sixth grade), and there is little improvement after that. That doesn't quite accord with the USC students performing better than the 11th-graders, so perhaps we need to know a little bit more about the evolution of strategic reasoning among older adolescents. However, in relation to younger children, Brocas and Carillo note that:
Existing research shows that by 7 years of age children may think ahead and form correct anticipations... Children have also been shown to develop inductive logic between the ages of 8 and 12...
Those are the sorts of skills that are used in developing level-k reasoning, so the mechanisms underlying the increase in strategic reasoning between ages 8 and 12 seem plausible. However, this clearly needs to be unpacked a bit more, and that would be a fruitful avenue for future research.
Overall, these two studies help us to understand a little bit more about how (and when) our reasoning in strategic games develops.
[HT: My colleague Steven Tucker for the Dufwenberg et al. study; Marginal Revolution for the Brocas and Carillo study]
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