Beauty Contest Game

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Beauty Contest Game

• Players: Two or more players.
• Strategies: Each player chooses a number $x_i\in[0,100]$.
• Rules: Players choose simultaneously before observing $\bar{x}$; the closest choice to $p\bar{x}$ wins.

Derivation (Iterative Deletion of Dominated Strategies)

Round 1

• If all choices are in $[0,100]$, it is a reasonable assumption that $\bar x \approx 50$.
• Since $\tfrac{1}{2}\bar{x}\le 50$, any strategy $x>50$ is dominated by choosing $\le 50$.

Round 2

• If all choices are in $[0,50]$, it is a reasonable assumption that $\bar x \approx 25$.
• Since $\tfrac{1}{2}\bar{x}\le 12.5$, any strategy $x>12.5$ is dominated by choosing $\le 12.5$.

Nash Equilibrium

Result:

Iterating yields shrinking upper bounds $50,25,12.5,\dots$, converging to $0$.

Under common knowledge of rationality, the unique Nash equilibrium is:

Insights

Insight:

Iterated reasoning drives the target toward zero when the multiplier is less than one.