# Introduction

The intersection of logic and decision theory has seen a revival in the past 30 years in multiple fields. In computer science, Boutilier [4] introduced the standard logical framework for qualitative decision theory. Another influential source is Halpern et.al [8]. In philosophy, interest in dynamic preference and epistemic logic has also spurred on the works of Van Benthem [9] and Liu [5], to cite two authors. Other fields such as economics and engineering have also contributed.

In this paper, we aim to give a logical formalization of temporal preferences. We will use Van Benthem’s standard modal preference model [9]. Van Benthem’s model is used to interpret game theoretic frameworks where the preference relation is indexed for each agent [9]. We should mention that Bienvenu [2] is an alternative method which has many advantages. Here, we consider an index over temporal preferences and an index for temporal location of states. It is necessary because we wish to distinguish between the following types of statements:

1) Time preferences: “I prefer having apples now rather than later.”

2) Temporal preferences: “I will prefer having apples rather than oranges.”

3) Combination of 1&2: “I will prefer having apples later rather than now.”

Namely, if we do not index preferences temporally, statements 1 and 3 are contradictions, although they may both simultaneously be true in some intuitive scenarios. Similarly, indexing over worlds is necessary if we wish to express discounting over time and patience effects.

# Semantics

#### Model Layout

Closely related to the model of Van Benthem et. al, our model is a 5-tuple of the form:

Where $W$ is the set of worlds, $T$ the set of temporal locations, $P$ the set of propositions and $V$ the valuation function. ${\precsim_{t\in T}}$ is the set of preferences the agent has at given temporal locations1.

Furthermore, worlds are denoted as $w_i^t\in W$; the subscript denotes the contents of the world and the superscript denotes its temporal location. Namely, the worlds $w_i^2$ and $w_i^3$ are the same world at different times, whereas $w_1^t$ and $w_2^t$ are different worlds at the same time. Lastly, the set of ${\precsim_{t\in T}}$ denotes the preference relations the agent has at different times. We take this to be a preorder on the set of worlds.

#### Syntax

Any well formed formula is recursively obtained as follows:

As per standard modal logic, we take the diamond operator to express the following:

Namely $\diamond_t\phi$ being true at $w_j^n$ implies that $\phi$ is true at a world which is weakly preferred at time $t$ to $w_j^n$. We take the $U$ operator to express universality2:

$U\phi \iff \forall w\in W$ we have $M,w\vDash \phi$

We denote the dual of $U$ and $\diamond_t$ by $E$ and $\Box_t$.

It is immediate that our semantics will have a standard axiomatization in S4 for which it is complete and sound since the index over worlds and preferences does not change the proof of soundness and completeness. The proof method, which involves the fusion of logic of S4 for each different modality, is entirely standard. Blackburn [3] is the authoritative reference on this topic.

# Temporal Decision Making

In this section we discuss how the rational agent should behave in our model. Two views are presented, one internal and one external. First we define the agent’s decision $d$ over $n$ periods as an n-dimensional vector $[w_{d1}^1,w_{d2}^2,…,w_{dn}^n]$. This represents their choice over worlds over time. Thus, a decision rule is a mapping from a set of temporal preferences and constraints on possible combinations of worlds to a decision.

In the temporal framework, there are two intuitive ways of formulating decision rules. The first is an internal one, using the notion of time preferences. The second is external and uses temporal preferences. We will show that under the right circumstances, the decision rules produced are identical.

#### Internal: Time Preference

If an agent’s decision rule depends on their time preference, it is said to be time dependent. The intuition is as follows: agents make present decisions based on what they presently want in the future and present. This view is internal, and the agent acts at each stage in an integrated and internal manner.

We can formulate the decision rule in an inductive manner as follows:

1) When $t=0$, given $\precsim_0$ and some constraint, we choose $w_{d0}^0$

2) When $t=n, n \geq 1$, given $\precsim_n$ and the updated constraint, which depends on $w_{dn-1}^{n-1}$, we choose $w_{dn}^n$.

3) So on and so forth.

The details of how a choice depend on the framework of the specified decision problem. Usually, we transform the preference order to its respective utility function and maximize it.

#### External: Temporal Preference

An agent making a decision based on their temporal preferences is one who weighs each preference ordering and decides like a social planner. This view is external since the decision rule is not part of the decision process. Therefore, the agent does not decide; rather, the decision has been made before the first period. We see that the type of agent that this decision rule models is much more “rational” as temporal location does not distort decision making. This does not model humans as well, but is better suited for machines.

Thus, the decision rule in its full generality is analogous to a social choice function which depends on temporal preferences rather than those of different individuals. Here we give a specific example:

1) Assign a weight $\beta_t$ to each temporal location.

2) Assign a utility function $u_t$ which reflects each preference relation $\precsim_t$.

3) Maximize $\sum_{t=0}^T \beta_t u_t(w_{dt}^t)$ given the constraint.

Further Characterization and Equivalence

In this section we will show that the two decision rules are identical in the most general case. First we need to define more formally the two types of decision rules.

The time preference decision rule is a set of decisions each made at time $t$ given the preferences of time $t$ and a constraint, which is given by previous decisions and the original constraint.\ Formally speaking, we can think of each decision at $t$ as a mapping from the set of preferences and set of possible allocations (constraints):

Where ${w_i}_t^n$ is a set of vectors from $t$ to $n$ (possible allocation/constraint) and $d$ is a decision which is itself a vector of world $t$ to $n$.

Now consider the following:

Instead of producing a decision, we simply examine one of the worlds of the decision, namely the one chosen for the current temporal location.

Similarly, let us define for some $w_j^t$ and constraint ${w_i}_t^n$, a mapping to the constraints for the agent in the next period.

We can now define the decision of the time preference as follows:

$D_1(\precsim_1,\{w_i\}_1^n)=w_1$ $D_2(\precsim_2, C_1(w_1,\{w_i\}_1^n))=w_2$ $\dots$ $D_n(\precsim_n, C_{n-1}(w_{n-1},C_{n-2}(w_{n-2},C(...C(w_1,\{w_i\}_1^n))...)= w_n$

Comparatively, the temporal preference decision rule is much easier to define:

We see that time preference decision rules are special cases of temporal preference decision rules. Furthermore, we can show, by construction, that every temporal preference decision rule has an equivalent time preference decision rule.

We can derive the results above because we have not imposed additional properties which would be natural to impose on decision rules. For instance, we would expect the $D_1,…,D_n$ of time preference decision rules to have similarity with each other. Alternatively, we might want the temporal preference decision rule to satisfy some intuitive properties, such as Pareto-Optimality.

# Conclusion

We maintain that time preferences are distinct from temporal preferences. Semantically, the difference is intuitively clear. We have also shown that they overlap in the model only when preferences are time consistent with some form of uniform change over time. Further, the result of equivalent decision rules is slightly misleading, since our decision rules would not capture the full intuition of the time/temporal preferences as they are without further characterizations.

### References

1. Arrow, Kenneth J. ”Extended sympathy and the possibility of social choice.” The American Economic Review (1977): 219-225.

2. Bienvenu, Meghyn, Christian Fritz, and Sheila A. McIlraith. ”Planning with Qualitative Temporal Preferences.” KR 6 (2006): 134-144.

3. Blackburn, P., van Benthem, J.F. and Wolter, F. eds., 2006. Handbook of modal logic (Vol.3). Elsevier.

4. Boutilier, Craig. ”Toward a logic for qualitative decision theory.” In Proceedings of the Fourth International Conference on Knowledge Representation and Reasoning (KR’94), pp.75-86. 1994.

5. Liu, Fenrong. Reasoning about preference dynamics. Vol. 354. Springer Science Business Media, 2011.

6. Gibbard, Allan. ”Manipulation of voting schemes: a general result.” Econometrica: journal of the Econometric Society (1973): 587-601.

7. Gul, Faruk, and Wolfgang Pesendorfer. ”Temptation and selfcontrol.” Econometrica 69, no.6 (2001): 1403-1435

8. Fagin, Ronald, Yoram Moses, Moshe Y. Vardi, and Joseph Y. Halpern. Reasoning about knowledge. MIT press, 2003

9. Van Benthem, Johan, Sieuwert Van Otterloo, and Olivier Roy. Preference logic, conditionals and solution concepts in games. ILLC, 2005.

### Notes

1. It is possible to define the set of temporal preferences fully with just one preorder, but we need to impose conditions such as time consistency and some form of uniform change over time. In this case, time preferences and temporal preferences are interdefineable. However, the conditions are often not realistic and in general we need to fully specify the preferences.

2. The $U$ operator and its dual are particularly useful to reduce preferences. Specifically, they translate preferences over worlds to preferences over propositions. For instance, the formula $U(\phi\rightarrow \Box_t\psi)$ denotes that all $\psi$ worlds are preferred to $\phi$ worlds at time $t$