# State dependent preferences vs state independent preferences in utility theory

I am working on changes in preferences and found papers on state-independent preference. What is the difference between state-dependent and state-independent preferences and utility functions? What are the assumptions change between the two? What is a textbook reference to understand the basics of two?

• Do you mean state-dependent expected utility? It would be helpful to narrow it down. Jan 11 at 12:26
• I read state-dependent utility functions actually in literature. Jan 14 at 16:50
• It would be helpful if you wrote which literature. Jan 14 at 17:31

Starting from the textbook, I would highly recommend any textbook for stochastic dynamic optimization. Then I would recommend you to get acquainted with markov chains, because it is relatively good introduction to how states might work. If you want something more from economics side, a good read might be anything on Savage framework. Either his original work (1954) or following works from different authors.

What concerns the difference between state-dependence and state-independence, it all comes from how states and preferences work. State of the world might be depicted as binary raining/sunny dichotomy. Ordering of (rational = transitive + complete) preferences is captured by the utility function. Then:

• State Dependent preferences: Ordering of preferences depends on the state of the world. For example, consumer who can buy ice cream or steak, might prefer steak over ice cream if it is raining, and ice cream over steak if it is sunny.
• State Independent preferences: This is the opposite. Your ordering of preferences is always the same, whatever happens. If you prefer steak over ice cream when it is sunny, you will prefer it also when it is raining.

Explaining this more robustly, you have a utility function $$U$$, for which it holds either $$U = U(\boldsymbol{x})$$ in case of state-independent preferences ($$\boldsymbol{x}$$ being choices) or $$U = U(\boldsymbol{x}, \boldsymbol{\psi})$$ in case of state-dependence ($$\boldsymbol{\psi}$$ being the states indicator).

What concerns some critical distinction between these two, state-dependence causes dynamic inconsistency by itself (stronger factor than discounting function which is not logarithmically linear).

• States in Markov chains and in decision theory have pretty much nothing to do with each other. Jan 12 at 19:30
• @MichaelGreinecker Would you not consider markov decision process as a prime example of connection between markov chains and decision theory? Jan 12 at 23:01
• I would see this as part of optimization theory, but I'm fine with seeing it as part of decision theory. But it has nothing to do with the "states of the world" Savage and others were talking about. Jan 12 at 23:07
• It is possible I misunderstood it. I always understood the Savage's states of the world as some random variable: Healthy/Sick ; Sunny/Raining; or even some quantitative continuous ones... That, however, seems to me as close as possible to the Markov interpretation of states. If I have some lottery (for example: health while insured), some realization of this random variable would change my ordering of preferences, which seems to be in harmony with Savage's framework, but to me it is also similar to the case in which my utility function depends on, for example, current mood... Jan 12 at 23:18
• A random variable is actually a function on a probability space, and the points of the probability space would be what the states are. But I don't think reading books on probability spaces would help in relation to conceptual discussions about states such as this, this, or this. Jan 12 at 23:46