Incomplete-Information Trading Game

• Content map: SMU H3 Game Theory Map

Setup

Definition:

Incomplete-Information Trading Game

• Players: Three players, Player 1, Player 2, and Nature, who independently draws wallet values.
• Strategies:
  • Nature draws $w_1$ for Player 1 and $w_2$ for Player 2.
  • Values are independent and uniformly distributed on $\Omega=\{5,6,7,\dots,55\}$.
  • Player 1 observes $w_1$ and Player 2 observes $w_2$ only, but both players know Nature’s distribution.
  • Each player chooses Keep or Swap after observing their own wallet.

Rules

• Start with each player privately observing an independently drawn wallet value; Players simultaneously choose Keep or Swap after observing their own wallet.
• The player who swaps only when it is profitable under beliefs maximises expected value.
• If both players choose Swap, they trade wallets; If at least one player chooses Keep, no trade occurs.
• Payoff equals final wallet value.
• A Nash equilibrium is a pair of strategies $(\sigma_1,\sigma_2)$ such that both strategies are mutual best responses.

Payoff Matrix

Derivation (Best Response Analysis)

Dominance argument

• Use iterated deletion of weakly dominated strategies.

• If $w=55$,

  • Keep gives payoff of exactly $55$.
  • Swap gives payoff of at most $55$.
  • Hence, Keep is never worse and sometimes better than Swap.

• If $w=54$,

  • Keep gives payoff of exactly $54$.
  • Swap gives payoff of at most $54$.
  • Hence, Keep is never worse and sometimes better than Swap.

• Continue iteratively down to $w=6$, resulting in the only non-unique action at $w=5$.

Cutoff strategy

• Suppose Player 2 uses the cutoff strategy

• If Player 1 swaps, then:
  • if $w_2\leq c_2$, trade occurs and Player 1 gets $w_2$,
  • if $w_2>c_2$, no trade occurs and Player 1 gets $w_1$.

• If Player 1 keeps, then:

Cutoff condition

• Keep is optimal if

Example cutoff

• For example, if $c_2=26$, then

• The cutoff is

• Thus Player 1’s best response is

Derivation (Nash Equilibrium)

Candidate cutoff strategies

• Let both players use cutoff strategies:

Best-response equations

• Best-response cutoff conditions are

• Solving gives

Cutoff equilibrium strategy

• Therefore the cutoff equilibrium strategy is

Always-keep strategy

• The always-keep strategy is

Equilibrium verification

• To verify $\sigma^2$ is a best response to $\sigma^2$:
  • If $w_i>5$, then $\sigma^1$ and $\sigma^2$ give the same payoff because no trade occurs.
  • If $w_i=5$, then under $\sigma^2$ trade occurs only when the opponent also has $5$, so payoff is always $5$.
  • If $w_i=5$, then under $\sigma^1$ payoff is also $5$.
• Hence $\sigma^2$ is a best response to $\sigma^2$.
• The same argument applies symmetrically to both players.
• Since $\sigma^1$ and $\sigma^2$ differ only at $w=5$, where the player receives payoff $5$ either way, the following are also equilibria:

Nash Equilibrium

Result:

The cutoff equilibrium is

where each player swaps only when $w=5$ and keeps otherwise.

The strategy pairs

are Nash equilibria.

Social Optimum

• Total payoff is always

• Swapping only redistributes wallet values between players.
• Ex post, both players weakly prefer swapping only when $w_1=w_2$.
• There is no utilitarian surplus gain from trade.

Insights

Insight:

• Willingness to swap signals a low wallet value.
• Higher types refuse to trade because the opponent’s willingness to trade is bad news.
• Trade occurs only at the lowest possible type.
• This is a classic adverse-selection or lemons-market outcome.