Optimal strategy in a single-agent choice problem under uncertainty - Economics Stack Exchange most recent 30 from economics.stackexchange.com 2019-12-08T23:33:33Z https://economics.stackexchange.com/feeds/question/29953 https://creativecommons.org/licenses/by-sa/4.0/rdf https://economics.stackexchange.com/q/29953 0 Optimal strategy in a single-agent choice problem under uncertainty user3285148 https://economics.stackexchange.com/users/1505 2019-06-25T20:51:23Z 2019-06-25T23:00:23Z <p>Consider the following single-agent choice problem under uncertainty. </p> <p>Let <span class="math-container">$V$</span> be the state of the world with support <span class="math-container">$\mathcal{V}$</span> and probability distribution <span class="math-container">$P_V\in \Delta(\mathcal{v})$</span>. First, let nature draw a realisation <span class="math-container">$v$</span> of <span class="math-container">$V$</span> from <span class="math-container">$P_V$</span>. Then, let the decision maker choose an action <span class="math-container">$y\in \mathcal{Y}$</span>, with <span class="math-container">$\mathcal{Y}$</span> finite, without observing <span class="math-container">$v$</span>. Upon the decision has been made, the decision maker gets a payoff <span class="math-container">$u(y,v)$</span>.</p> <p>For example, suppose that <span class="math-container">$\mathcal{Y}\equiv \{1,2,3\}$</span>. <span class="math-container">$V$</span> is a <span class="math-container">$3\times 1$</span> random vector, <span class="math-container">$V\equiv (V_1,V_2,V_3)$</span>. <span class="math-container">$P_V$</span> is the 3-variate standard normal distribution. <span class="math-container">$u(y,v)\equiv v_y$</span>. </p> <p><strong>What is the definition of an optimal strategy for the decision maker in this setting?</strong> </p> <p>I'm thinking about using "a sort of" Bayesian Nash equilibrium for a single-agent setting, i.e., an optimal strategy is <span class="math-container">$P_Y\in \Delta(\mathcal{Y})$</span> such that, <span class="math-container">$\forall y\in \mathcal{Y}$</span> such that <span class="math-container">$P_Y(y)&gt;0$</span> and <span class="math-container">$\forall \tilde{y}\neq y$</span>, we have that <span class="math-container">$$\sum_{v\in \mathcal{V}} u_i(y,v)P_V(v)\geq \sum_{v\in \mathcal{V}} u_i(\tilde{y},v)P_V(v)$$</span> that is, in my example, <span class="math-container">$$\sum_{v\in \mathcal{V}} (v_y -v_{\tilde{y}}) P_V(v)\geq 0$$</span></p> <p>But maybe a <strong>pure</strong> strategy is what people use? </p> <p>Is <strong>existence and uniqueness</strong> obvious (at least in my example with a normal distribution)? </p> <p>Could you also provide a reference discussing <strong>definition, existence, multiplicity</strong>?</p> https://economics.stackexchange.com/questions/29953/-/29958#29958 1 Answer by Regio for Optimal strategy in a single-agent choice problem under uncertainty Regio https://economics.stackexchange.com/users/21827 2019-06-25T23:00:23Z 2019-06-25T23:00:23Z <p>Your problem can be simply expressed as <span class="math-container">$$\arg\max_{y\in\mathcal{Y}}\sum_{v\in\mathcal{V}}u_i(y,v)P_V(v)$$</span> Note that the subscript in <span class="math-container">$u_i$</span> is un-necessary. Furthermore, this is not a game so "mixed strategies" are only a solution if the maximum is not unique, but you should not worry about them (more on that later). </p> <p>First, note that the maximization problem is equivalent to the inequality you present. Presenting it as a maximization problem is more akin to how you solve a single agent problem, and circumvents the need to specify the quantifiers. However, strictly speaking, it only characterizes "pure strategies", (a better term would be deterministic actions), but this is, really, without loss of generality. A probabilistic action is justified only when the maximizer is not unique, in which case the <span class="math-container">$\arg\max$</span> operator will already be giving you a set, and any mixture between the maximizers in that set would be optimal. </p> <p>Besides being more clear, another benefit from presenting it as a maximization problem is that you can use standard theorems to guarantee existence and uniqueness, for example, if <span class="math-container">$\mathcal{Y}$</span> is finite then <span class="math-container">$\sum u(y,v)P_V(v)$</span> is continuous in <span class="math-container">$y$</span> (using the discrete metric) and Weierstrass ensures the existence of a solution. In general you need a <span class="math-container">$\mathcal{Y}$</span> to be compact and <span class="math-container">$\sum u(y,v)P_V(v)$</span> to be continuous in <span class="math-container">$y$</span>. Uniqueness is a bit more tricky, but if <span class="math-container">$\sum u(y,v)P_V(v)$</span> is strictly concave with respect to <span class="math-container">$y$</span>, then uniqueness is guarantied. </p> <p>In your particular example given the finiteness of the action space, existence is trivial, but not uniqueness. Knowing that <span class="math-container">$P_V$</span> is normal is not enough since we don't know anything about the shape of <span class="math-container">$u(\cdot)$</span>. </p>