Imputations

Game Theory · Axiom Academy

Understanding how to allocate value fairly in cooperative games An imputation is a vector x = (x₁, x₂, ..., xₙ) where xᵢ represents the payoff to player i . The first requirement is efficiency : the sum of all payoffs must equal the total value the grand coalition can create. The second requirement is individual rationality : each player must receive at least as much as they could get by acting alone. If a player would get more by going solo, they have no incentive to join the coalition. The collection of all allocations satisfying both efficiency and individual rationality forms the imputation set . This set contains all feasible, rational ways to divide the value among players. 4. Valid vs. Invalid Imputations Let's examine concrete examples with three players where v(N) = 90 and each player can earn 10 alone: v( 1 ) = v( 2 ) = v( 3 ) = 10. We'll see which allocations qualify as imputations and which don't. 5. Why Individual Rationality Matters Individual rationality isn't just a mathematical constraint—it captures a fundamental principle of voluntary cooperation. Without it, rational players would reject the coalition, making cooperation impossible.

This is the written version of the interactive lesson above. See the full Game Theory course.