# Expected Utility Transformations

This lecture looks at a rare time that weak dominance can actually be useful!

Takeaway Points

1. If an individual’s preferences follow the four covered axioms (completeness, transitivity, independence, and continuity), we can represent those preferences with a numerical utility function.
2. However, the utility function is not unique. Suppose a set of utilities accurately reflects a player’s preferences over outcomes. Let u be the individual’s utility for an outcome, a > 0, and b be any real number. Then converting all utilities by the transformation au + b maintains identical preferences.
3. Consequently, if we take a player’s utilities from a game and convert them all by au + b, the equilibria will be the same across the games. This remains true even if we change both players’ utilities, using a different a and b for each player.
4. Interestingly, the au + b transformation is the only transformation that preserves equilibria in this manner.

Back to Game Theory 101