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# Hide & Seek

## Game Description

Each player makes two decisions. The first is to choose under which of four cups to Hide a coin, then choose under which of four cups to look for (Seek) a coin. A player's Total Payoff equals her Hiding Payoff plus her Seeking Payoff. Her Hiding Payoff equals the number in her group who do not choose to seek under the cup she chose for Hide. Her Seeking Payoff equals the number in her group who chose to hide under the cup she chose for Seek. (Just like the childhood game, one wants to hide where others do not seek, and seek where others choose to hide.)

### Learning Objective 1: Mixed Strategies

In the unique Nash equilibrium, each player uniformly randomizes across locations when both seeking and hiding (i.e., chooses each cup with probability equal to 0.25).

### Learning Objective 2: Choice Salience and Strategic Thought

Framing and context may alter the salience of different choices. As a result, they may affect non-equilibrium reasoning and actual choices even when the context is not (directly) payoff relevant.

## Brief Instructions

Especially for the one-period game, most instructors opt for large group sizes. To have all of your students in one group, make sure Group Size is larger than the number of students. A period ends when all students in a group submit both decisions. In addition, the game (and thus the current period) ends when either the game reaches its Duration or you click Finish.

The game has a unique Nash equilibrium: each player uniformly randomizes across locations when both seeking and hiding (i.e., chooses each cup with probability equal to 0.25). However, the framing of the decision makes for a richer strategic environment.

The second cup, which is the only red cup, is clearly the most salient, or focal, cup. Arguably, the cups at the end (cups one and four) follow in salience. This leaves the third cup as the least salient option.

To see how framing might affect decision making, start with hiding, the student's first decision. At a first level of strategic thinking, a student might choose to hide the coin in the least salient location: cup 3. However, a hider might might engage in an extra step of strategic thinking: if hiders are likely to choose the least salient choice, then seekers would be wise to look under cup 3. Therefore, I am going to hide under the most salient cup, the red cup! Of course, some seekers are going to anticipate this (because hiders think that seekers will not look under the most salient cup, I am going to seek under the most salient cup) and seek under the red cup.

Cup 1 Cup 2 Cup 3 Cup 4
Hide (n=53) 9% 36% 40% 15%
Seek (n=62) 13% 31% 45% 11

In the table above, we reproduce the results from Rubinstein, Tversky, and Dana (1996)1 as reported in Crawford and Iriberri (1997)2. We expect that you will get similar results. The results are consistent with story above. The most common choice—for both hiding and seeking—is the least salient option (cup 3), consistent with a low level of strategic reasoning. The second most common choice —for both hiding and seeking—is the most salient option (red cup 2), consistent with a next level of strategic reasoning.

## Key Treatment Variations

Repetition may move play closer to the Nash equilibrium prediction. You have a couple of options:

1. After each one-period game, click either Copy or Replay.
2. Play a multi-period game by changing Minimum Periods in the Repetition panel. A group proceeds to the next period only when all group members have made both choices, meaning that incomplete groups (number of players less than group size) cannot proceed to subsequent periods. While you can complete a group by clicking Add Robots in its listing in your game monitor, these robots will play with the frequencies described in the table above.

## Results

The first set of charts (Figure 1) shows the distribution of hide choices, the distribution of seek choices, along with the Nash prediction (group size divided by four) for both. As argued above, you will likely see cups 2 and 3 played more frequently, and 1 and 4 less frequently.

The final chart (Figure 2) displays the frequency of each action combination. If students independently randomized each decision, all frequencies would be (approximately) equal to group size, divided by 16 actions. Because we ask the "match" question second, you are likely to see many observations on the diagonal: the student who has just deduced that cup 3 would make an ideal hiding place will likely now choose it as a likely hiding place.

This chart also displays the payoff consequences of each action combination. Recall that a player's Total Score equals the number of times she successfully hid plus the number of times she successfully sought. In order to remove the effect of group size, we calculate (and present to students) an Overall Score equal to a students total score divided by the group size and multiplied by 100. As the Total Score ranges from 0 to twice the group size and in equilibrium is equal (in expectation) to the group size, the Overall Score ranges from 0 to 200 with an equilibrium benchmark of 100.

In the Nash equilibrium with large groups, there would be little variation across choice combination in either payoffs or frequency of play. With your students, you are likely to see actual payoffs that vary greatly from 100 (in the data above, overall scores range from 64 to 129) and frequencies of play departing from the benchmark of group size divided by 16.

In the results we display, highest scores are earned by those who hide where others do not seek (cups (rows) 1 and 2) and seek where others hide (cup (column) 3).

1. Rubinstein, Ariel, Amos Tversky, and Dana Heller. 1996. "Naïve Strategies in Competitive Games," In Understanding Strategic Interaction—Essays in Honor of Reinhard Selten, ed. Wulf Albers, Werner Güth, Peter Hammerstein, Benny Moldovanu, and Eric van Damme, 394–402. Berlin: Springer-Verlag.
2. Crawford, Vincent P., and Nagore Iriberri. "Fatal Attraction: Salience, Naivete, and Sophistication in Experimental Hide-and-Seek Games." The American Economic Review 97, no. 5 (2007): 1731–1750.