Experiment 12 Expectations
I Think You Think…
Pick a number, guess what others expect, and see how a group learns across repeated rounds.
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Start with “Work It Out, Step by Step” below. The optional expert section explains its symbols as you go. For more examples, use the plain-language math guide.
1. Make a Prediction
How Low Will Everyone Go?
You and 19 simulated players each choose a number from 0 to 100. Add all 20 numbers and divide by 20 to find the average. The target is two-thirds of that average. Whoever gets closest shares a 100-point prize. Your own number counts in the average.
2. Make Your Choice
3. Reveal the Incentives
Think About the People at This Table
A reasoning depth counts how many times a player takes two-thirds, starting from a guess that the average will be 50. Depth 0 chooses 50. Depth 1 chooses about 33.33. Depth 2 chooses about 22.22. Depth 3 chooses about 14.81. Depth 4 chooses about 9.88. These are programmed rules of thumb, not intelligence scores.
| Round | Your number | Group average | Target | Your distance | Your prize |
|---|
Try 0 against players who all start at 50. Then reset and try a number near 33. Thinking through more steps does not guarantee a win against the actual people in front of you.
4. Change One Assumption
Change the Starting Beliefs or the Learning Speed
Changing a setting starts a new run. A run lasts at most 30 rounds. Zero percent keeps the simulated players’ numbers fixed. One hundred percent makes each choose the previous target next time. There is no randomness, so replaying the same choices gives the same results.
Work It Out, Step by Step
Higher-order beliefs are thoughts about other people’s thoughts. “I think the average will be 50” is one belief. “I think everyone else expects 50 and will choose two-thirds of it” adds another layer.
- Find the average. Suppose all 19 other players choose 50 and you choose 30. Their numbers add to 950. Adding yours gives 980. Divide by 20 players: the average is 49.
- Find two-thirds. Divide 49 by 3, then multiply by 2. The target is about 32.67.\[T=\frac{2}{3}\times49\approx32.67.\]
In words: “The target equals two-thirds times 49, which is approximately 32.67.” The letter T names the target. The symbol \(\approx\) means “approximately equal to”; we rounded the answer.
- Compare distances. Your 30 is about 2.67 away. Each other player’s 50 is about 17.33 away. You get all 100 prize points in this example.
- Watch learning change the next round. At 50% learning, a player at 50 moves halfway toward 32.67. Half of 50 is 25, and half of 32.67 is about 16.33. Add them to get a next choice of about 41.33.
The lowest possible number is 0. If everyone chooses 0, the target is also 0 and everyone shares the prize. That is a Nash equilibrium: nobody can earn more by changing only their own number. It does not mean 0 wins against every starting population.
For experts: formal model and assumptions
From Guesses to the Target
- \(x_i\), \(N\), and \(T\)In words: “player i’s number,” “number of players,” and “target.”
- The small lower i is a player label, called a subscript. There are \(N=20\) players. Every choice lies from 0 through 100.
- \(\sum_{i=1}^{N}x_i\)In words: “add the choices from player 1 through player N.”
- The large Greek letter sigma means to add a list. Dividing that sum by the number of players gives the average. See the guide to adding a list.
In words: “Add every choice, divide by the number of players, and take two-thirds.” The fraction combines those two steps. The prize is split equally among choices with the smallest absolute distance from T. An absolute distance is the gap without a negative sign: 30 is 2.67 away from 32.67.
The Programmed Learning Rule
- \(x_{i,t}\), \(x_{i,t+1}\), and \(T_t\)In words: “player i’s current number,” “that player’s next number,” and “this round’s target.”
- The small t labels a round. Adding 1 moves to the next round. The comma separates the player label from the round label.
- \(\alpha\)In words: “alpha,” the learning fraction.
- Alpha is the Greek letter used here for how far a simulated player moves toward the target. A setting of 50% means \(\alpha=0.5\). Then \(1-\alpha\), read “one minus alpha,” is the fraction of the old guess retained. See rounds and updates.
In words: “The next guess is the retained part of the old guess plus the learning part of the latest target.” Letters placed next to parentheses mean multiply. This is an adaptive simulation rule, not a claim that the simulated players calculate an optimal response.
The mixed starting population has 3 players at depth 0, 5 at depth 1, 5 at depth 2, 3 at depth 3, and 3 at depth 4. A depth-k starting guess is \(50(2/3)^k\). In words: “Start at 50 and multiply by two-thirds k times.” The raised k is an exponent, which counts repeated multiplication. These starting rules of thumb do not correct for a player’s effect on the average. The scoring rule does include that effect. The 100-point prize is conserved each round, including ties.
Concept reference: Vincent Crawford’s university lecture notes on level-k reasoning and guessing games. This toy population is not fitted to those experiments or to human behavior.
5. Take It Somewhere Else
What Will Other Teams Budget?
A planning team predicts the budget requests other teams will submit. Those teams are themselves trying to anticipate the planning team. Would adding another layer of reasoning necessarily make the forecast better?
Reveal the reasoning
Only if that extra layer resembles how the other teams actually decide. A sophisticated forecast can fail when it assumes everyone else reasons the same way. Gather evidence about their rules and update the forecast when their behavior changes.
A Model Is a Place to Start
These small models make the incentives visible. Their results follow from their stated rules; they are not forecasts of how every person or organization behaves. A simulated strategy is a rule, not a personality.
Scenario links save the controls and random seed. To reproduce an interactive run, make the same choices in the same order. Changing a setting restarts the experiment.