Game Theory 101 (#51): Expected Utility Transformations

[@liibernovski]

Super-clear explanation!

[@robinwang6399]

I think we might need to restrict a&b to be finite. Any limit taking a or b to infinity will result in a equal sign, because ab are both real positive numbers and have the same cardinality.

[@InventiveHarvest]

I could be incorrect here, but I think you can have an order preserving function if it is piece-wise. For example: function of utility equals utility squared if utility is positive and negative utility squared if utility is not positive. -2 < -1/4 < 0 < 1/4 < 2/3 < 2 = -4 < -1/8 < 0 < 1/8 < 4/9 < 4.

[@Gametheory101]

+Inventive Harvest That preserves the ordering but will mess with the mixed strategies. You can try it with a game that has an MSNE, or just think of the preference over 1/4 with certainty and a lottery between 0 and 2 and compare that to the preference over 1/16 with certainty and a lottery between 0 and 4. There are infinitely many other ways to preserve the ordering (for example, u^n, for all n odd), but only positive affine transformations preserve the preferences over lotteries as well.

[@PunmasterSTP]

This was a fine video; thanks for sharing! 😎

[@carycrichlow]

Sorry this has nothing to do with this video but I had a question about your thoughts on Nash equilibrium and if we could use it in the context of choosing to be on welfare or not? What types of conditions would lead us to be indifferent about being on it or not? Not looking for answers but wondering if its a valid question? I'd appreciate your thoughts on the matter. Thanks!!

[@rohanvaswani9418]

In order to have the same nash equilibrium for both games, don't you have to do a positive affine transformation for both players? otherwise if you multiply all of player two payoffs by 100 for example and leave 1 unchanged then the nash equilibrium would surely change?

[@flippo1473]

No, it wouldn't change the NE since utilities are compared for each player respectively.

[@ashbrmi]

You did fine on the French pronounciations, but you didn't get the ¨Morgenstern¨ quite right. Check with a German speaker.