# Shouldn't the utility function of two-player zero-sum games be in the range $[-1, 1]$?

In Appendix B of MuZero, they say

In two-player zero-sum games the value functions are assumed to be bounded within the $$[0, 1]$$ interval.

I'm confused about the boundary: Shouldn't the value/utility function be in the range of [-1,1] for two-player zero-sum games?

## 1 Answer

it can be either. If you consider the lack of reward as "penalty" then getting 0 reward is bad.

if you use a value estimator through a neural network, the range of rewards will dictate the squashing function you use for the output layer

• Hi, @nikos if we use [0,1] as the bound for the value functions, then how can the game be zero-sum? – Maybe May 6 '20 at 1:07
• The utility function is valued from the perspective of ONE of the 2 players, so it definitely shouldn't sum to zero! If you train it properly, it should give a larger percentage (probability) to the AI player winning. You should regard "zero sum" in the sense one wins and the other loses – nikos May 7 '20 at 10:50
• Hi @nikos. The game will return either $+1$ or $-1$ to the agent to indicate whether it wins or not. I get that the agent is unlikely to take actions that cause it to lose, but I cannot see any guarantee that restricts the value function to positive. For example, if the agent is in a state where there is no way to win if its opponent acts optimally. Shouldn't the value function of that state be negative in that case? – Maybe May 8 '20 at 11:48
• I haven't read the particular paper that you refer to, but what I've been trying to tell you all along is that the actual range of rewards doesn't matter. If you want [-1,1], that's fine, and [0,1] is also fine, as well as [-1000,1000]. You just have to be consistent – nikos May 12 '20 at 11:04
• I got that, but this range is not an arbitrary choice; it affects the value of pUCT, which is important to MCTS. – Maybe May 13 '20 at 1:00