# Information-Theoretic Bounded Rationality and ε-Optimality

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{j}∈ each of which can occur with a respective probability P(o

_{j}) where j = 1, …, N. We can imagine a lottery as a roulette wheel or a gamble where we obtain a prize o

_{j}with probability P(o

_{j}) that has a subjective utility U(o

_{j}) for the decision maker. The compound value of the lottery can then be determined by the expected utility

**E**[U] = ∑

_{j}P(o

_{j})U(o

_{j}), which is commonly used as the standard performance criterion in decision making. The concept of expected utility was first axiomatized by Neumann and Morgenstern [8]. In their axiomatic system, Neumann and Morgenstern [8] define a binary preference relation ≻ over the set of probability distributions ℘ defined over the set of outcomes . If (and only if) this binary relation satisfies the axioms of completeness, transitivity, continuity and independence, then there exists a function U : ↦ ℝ, such that:

_{j}|a

_{i}), where a

_{i}∈ is an action that leads to consequence o

_{j}with probability P(o

_{j}|a

_{i}). The decision maker can assess the expected utility of each action as

**E**[U|a

_{i}] = ∑

_{j}P(o

_{j}|a

_{i})U(o

_{j}). Thus, the probabilistic model of the world defines a set of M different lotteries indexed by a

_{i}, where i = 1, …, M. The decision maker can compare the expected utilities of all the lotteries and choose the one with the highest expected utility, such that:

_{max}, but for any action from a set of permissible actions whose expected utility deviates at most by ε > 0 from the optimal expected utility of a

_{max}, such that:

## 2. Methods

_{i}) reflecting this uncertainty. Information-theoretic models of bounded rational decision making quantify the cost of information-processing by entropic measures of information [15–17,31–35] and are closely related to softmax-choice rules that have been extensively studied in the psychological and econometric literature, but also in the literature on reinforcement learning and game theory [36–42]. In [31–34], Ortega and Braun discuss an information-theoretic model of bounded rational decision making where information processing costs are quantified by the relative entropy with the idea that information processing costs can then be measured with respect to changes in the choice strategy P(a

_{i}).

_{0}(a

_{i}). This could include the uniform distribution over a

_{i}as a special case, if the decision maker has no prior preferences between different actions. Next, this decision maker is exposed to a utility function V(a

_{i}), which includes the case of V(a

_{i}) =

**E**[U|a

_{i}], implying that the decision maker does not have to compute the expectation values, but the expectation values are simply given. Ideally, the decision maker will arrive at the new distribution P(a

_{i}) = δ

_{ai}

_{,}

_{amax}. The underlying computation can be imagined as a search process that reduces the uncertainty over the action by D

_{KL}[P||P

_{0}] = ∑

_{i}P(a

_{i}) log [P(a

_{i})/P

_{0}(a

_{i})]. In general, such a search is costly, and the decision maker might not be able to afford such a stark reduction in uncertainty. Assuming a price 1/α for 1bit of information gain, we can then design a bounded optimal decision maker that trades off gains in utility resulting from changes in P(a

_{i}) against the search costs that these changes imply, such that, overall, the decision maker optimizes a free energy difference in utility gains and information costs:

_{P̃}ΔF[P̃] is the equilibrium distribution:

_{1}[P̃] − F

_{0}, with the free energies:

_{0}(a

_{i}) = exp (α(Φ

_{0}(a

_{i}) − F

_{0})) and V(a

_{i}) = Φ

_{1}(a

_{i}) − Φ

_{0}(a

_{i}). Hence, the utility function V(a

_{i}) expresses changes in value Φ, that are gains or losses with respect to the status quo. In the case of inference, the utility function is given by a negative log-likelihood and measures informational surprise. The temperature parameter corresponds then to a precision parameter in exponential family distributions. Casting the problem of acting as an inference problem has been previously discussed in [43–48]. The certainty-equivalent value V

_{CE}under strategy P can be determined from the same variational principle:

_{i}) are not simply given, the decision maker has to compute the expectation values herself from the prior P

_{0}(o

_{j}|a

_{i}) and the utility U(o

_{j}), such that search costs have to be considered both for a

_{i}and o

_{j}. The variational problem can then be formulated as a nested expression [32,34,49]:

_{i}) is much cheaper than the calculation of the optimal action, then the price 1/β should be much lower than 1/α, such that α ≫ β, implying that we can simply obtain samples from P

_{0}(o

_{j}|a

_{i}) for our computation of the expectation, but that it is much more difficult to compute a

_{i}, because we cannot simply rely on our prior P

_{0}(a

_{i}). The two-part solution to the nested variational problem is given by:

_{β}(a

_{i}) = ∑

_{j}P

_{0}(o

_{j}|a

_{i}) exp (βU(o

_{j})) and:

_{i}), but we are only able to sample from the distribution P

_{0}(a

_{i}). In this scheme, we generate a sample a

_{i}~ P

_{0}(a

_{i}) and then accept the sample if:

_{i}V(a

_{i}). Otherwise, the sample is rejected. The efficiency of the sampling process depends on how many samples we will need on average from P

_{0}to obtain one sample from P. This average number of samples from P

_{0}needed for one sample of P is given by the mean of a geometric distribution:

_{max}= arg max V(x) and T > max

_{i}U(a

_{i}).

_{i}~ P

_{0}(a

_{i}) if it fulfils the criterion:

_{β}(a

_{i})/e

^{βT}can be interpreted as an acceptance probability; in this case, the acceptance probability of θ ~ P

_{0}(θ). Thus, in order to accept one sample from x, we need to accept ${\scriptstyle \frac{\alpha}{\beta}}$ consecutive samples of θ, with acceptance criterion:

## 3. Results

#### **Theorem 1** (ε-Optimality)**.**

_{max}= max

_{i}

**E**[U|a

_{i}] of the perfectly rational decision maker, such that:

#### Proof

_{CE}under the bounded rational strategy P(a

_{i}) is given by:

_{0}(a

_{i}) = 1/M that the ε-bound is given by ε = 1/α log M. Conversely, given an ε > 0, there exists an $\overline{\alpha}={\scriptstyle \frac{\text{log\hspace{0.17em}}M}{\varepsilon}}$, such that for α ≥ ᾱ, any decision taken yields a utility within epsilon of the optimum.

_{i}) =

**E**[U|a

_{i}], such a decision maker optimizes the “distorted” certainty-equivalent value:

_{β}(a

_{i}) from Equation (7). Only for β → 0, the expectation value Ṽ(a

_{i}) →

**E**[U|a

_{i}] is retained. Due to ${\scriptstyle \frac{1}{\beta}}\hspace{0.17em}\text{log\hspace{0.17em}}{Z}_{\beta}({a}_{i})\ge \mathbf{E}[U\mid {a}_{i}]$, such a decision maker with positive β will overestimate the certainty-equivalent value for sub-optimal actions a

_{i}. For small β ≪ 1, the certainty-equivalent value can be approximated by a Taylor expansion in β:

^{2}) are higher-order cumulants that can be neglected. Due to Theorem 1, we have:

_{i}) are very similar in magnitude, it requires a high rationality parameter α to differentiate between them. A tighter ε-bound in α can be given, if we assume that there is an interval V(a

_{i}) ∈ [V

_{min}; V

_{max}] and that all the utilities are discriminable by at least one “utile”, such that for any choice a

_{i}and a

_{k}, we have |V(a

_{i})−V(a

_{k})| ≥ 1, which is the case, for example, when utilities reflect rank.

#### **Theorem 2** (ε-Optimality for rank utilities)**.**

_{0}(a

_{i}) = 1/M, bounded (expected) utilities V(a

_{i}) ∈ [V

_{min}; V

_{max}] for all i and |V(a

_{i}) − V(a

_{k})| ≥ 1 for every pair (i, k), one can bound the expected performance of this decision maker from below within an ε-neighborhood of the optimal performance V

_{max}= max

_{i}

**E**[U|a

_{i}] of the perfectly rational decision maker, such that:

#### Proof

_{i}) derived from Equation (4) under uniform prior P

_{0}(a

_{i}) = 1/M as:

_{i}) − V(a

_{k})| ≥ 1 ∀i, k and the limit properties of the geometric series. Therefore, we have:

_{min}; V

_{max}] = [V

_{min}; V

_{min}+M] that the performance bound is given by ∑

_{i}P(a

_{i})V(a

_{i}) ≥ V

_{max}− e

^{−}

^{α}M. Conversely, given an ε > 0, there exists an $\overline{\alpha}=\text{log}{\scriptstyle \frac{{V}_{max}-{V}_{min}}{\varepsilon}}$, such that for α ≥ ᾱ, any decision made yields a utility within epsilon of the optimum.

## 4. Adversarial Environments

_{i}∈ with (expected) utility V(a

_{i}) =

**E**[U|a

_{i}].

#### 4.1. Unknown Action Set

_{i}) over actions a

_{i}∈ , and then, the environment chooses a subset ∈ ℘( )\{} of permissible actions, where ℘( ) denotes the powerset. All actions that are not part of the subset are eliminated. Finally, the action a

_{i}is randomly determined from the set of permissible actions with their renormalized probabilities. The problem is to find the betting probability P(a

_{i}) such that we maximize our expected return; however, the expectation has to be taken over the unknown subset capriciously chosen by the opponent. This models a decision maker, who has to choose a generic hedging strategy by allocating resources to different alternatives, but where the rules of the game are only fully revealed after the choice is made. Formally, we want to choose the probability P(a

_{i}), such that the conditional expectation

**E**[V(a

_{i})| ] is as large as possible. Unsurprisingly, we cannot provide a deterministic optimal solution P(a

_{i}) = δ(a

_{i}−a*), since the environment could always eliminate a*. However, if we allow ourselves an arbitrarily small, non-zero performance loss ε > 0, then there is a way to assign probabilities P(a

_{i}), such that the conditional expectation is almost equal to the optimum, i.e., to the highest utility in the subset chosen by the opponent. This is precisely the result of the following theorem.

#### **Theorem 3** (ε-Optimality in adversarial environments)**.**

#### Proof

#### 4.2. Unknown Utility

_{0}(a

_{i}) and the environment subsequently chooses V(a

_{i}) in an arbitrary fashion, such that, in general, the choice of V(a

_{i}) may depend on P

_{0}(a

_{i}). Once the V(a

_{i}) are revealed, the decision maker updates the choice strategy according to Equation (4). Importantly, the new distribution P(a

_{i}) is not used as a choice strategy to choose between the different V(a

_{i}) as in the previous theorems, but is only used in a later choice with new, yet unknown utilities. If we denote the trial number or time step by t and assume a trial-by-trial update:

_{t}(a

_{i}) are bounded in each time step to lie within the unit interval, that is V

_{t}(a

_{i}) ∈ [0; 1], then the expected performance of the decision maker can be bounded from below by:

_{t}(a

_{i}) in each time step [52,53]. In this case, the decision maker chooses between i different options with probability p

_{i}(t) = w

_{i}(t)/∑

_{j}w

_{j}(t), where the weights w

_{i}(t) are updated according to:

_{i}(t) is the utility of option i at time t. It is straightforward to see that a bounded rational decision maker following Equation (4) is hedging, when acting according to P

_{t}(a

_{i}) before receiving feedback V

_{i}(t); that is, the bounded rational decision maker has a delay of one time step, as it is the distribution P

_{t}

_{+1}(a

_{i}) that is bounded optimal for the utility V

_{i}(t) under the prior P

_{t}(a

_{i}).

## 5. Discussion and Conclusion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Braun, D.A.; Ortega, P.A. Information-Theoretic Bounded Rationality and ε-Optimality. *Entropy* **2014**, *16*, 4662-4676.
https://doi.org/10.3390/e16084662

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Braun DA, Ortega PA. Information-Theoretic Bounded Rationality and ε-Optimality. *Entropy*. 2014; 16(8):4662-4676.
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Braun, Daniel A., and Pedro A. Ortega. 2014. "Information-Theoretic Bounded Rationality and ε-Optimality" *Entropy* 16, no. 8: 4662-4676.
https://doi.org/10.3390/e16084662