Three-Player Gift Exchange Game

Optimal strategy in a sequential gift-exchange game

This easy probability problem tests your ability to reason backwards through a multi-stage sequential game and make a decision under uncertainty. It combines conditional expectation with game-theoretic thinking: you must decide whether to take a known value or gamble on a new draw, while anticipating what Player 3 will do after observing both gifts.

The core challenge is to model Player 3's optimal behaviour, then use that to compute your own expected payoff under each of your choices. You'll need to condition on the value you observe from Player 1, consider the probability distribution of future draws, and account for the risk that a high-value gift you steal may be taken from you. A clean solution explicitly lays out the decision threshold and why it exists.

• Backward induction in sequential games
• Conditional expectation given observed information
• Comparing known outcomes to random draws
• Equilibrium reasoning under perfect information