# Can von Neumann Morgenstern utility be negative?

I saw this question in stackexchange that a utility function can take negative values.

I did not read any thing from the axioms that utility cannot be negative. It states they must be in the set of real numbers with positive affine transformation, however does negative utility indicate the consumer is irrational? My initial thought is yes, they can be negative because it is the relative preferences that matter not the actual values. But VNM is cardinal which means values are important so maybe a rational person would not use negative preferences. If he was rational and truly disliked a good, the utility would be zero, correct? If I like apples more than bananas, I would be less likely to say the utility of bananas for me is -10 and the utility of apples is -5. Giving a minimum value of zero is more likely: the utility of banana is 0 and the utility of apples is 5.

The VNM utilities in standard micro and game theory textbooks that I saw have a minimum of zero so I was not sure if negative value utilities are allowed if a person is rational or if it is just for simple demonstrations.

• "I would be less likely to say the utility of bananas for me is -10 and the utility of apples is -5. Giving a minimum value of zero is more likely: the utility of banana is 0 and the utility of apples is 5." This depends entirely on your personal taste, $-10$ and $-5$ describe the exact same preferences as $0$ and $5$, utilities are not assumed to convey any additional information in this case. Aug 5, 2023 at 6:40

vNM utilities admit affine transformations: if $$U(\cdot)$$ is a vNM utility function over lotteries, then the function $$V(\cdot)$$ defined as $$V(\cdot) = aU(\cdot) + b$$ where $$a\in\mathbb{R}_{++}$$ and $$b\in\mathbb{R}$$ will also be a vNM utility function, describing the same preferences as $$U(\cdot)$$. If you make $$b$$ large enough, $$V(\cdot)$$ will take negative values sometimes.
I saw treatments of the prisoner's dilemma with non-positive payoffs/utilites for the pure strategy profiles: $$\begin{array} {cc} \\&C&D\\ C&-1,-1&-10,0\\ D&0,-10&-5,-5 \end{array}$$ If one were to calculate expected payoffs/vNM utilites for the game, they too would be non-positive. Arguably this is because here the players get something 'bad' at the end of the game; but even this is arbitrary. One could get a strategic equivalent of this game with positive payoffs by adding +20 to all payoffs, an affine transformation of both payoff/utility functions.