Questions tagged [decision-theory]

the mathematical study of strategies for optimal decision-making between options involving different risks or expectations of gain or loss depending on the outcome.

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Can the topological assumption in Debreu's representation theorem of cardinal utility be altered from "connected separable" to "second countable"?

Theorem (Debreu 1959 page 9, 10) Let $X$ be connected separable topological space endowed with product topology. If $\succsim$ is independent and at least three factors are essential, then there exist ...
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Does Debreu's representation theorem of ordinal utility require Hausdorff topology?

By Debreu's theorem of ordinal utility, any continuous weak order on $X$ is represented with a continuous utility function, if $X$ is a second countable or connected separable topological space. My ...
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How do we have no–envy for the Final Target Score here in assessing the consequences of individual changes on game dynamics and appeal?

Assume that a new Chess-variant designed to increase the level of competition throughout the game, provide additional excitement at the finish: "A Final Target Point will be set." The Final ...
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How did econometricians justify the use of $EU$ instead of $EU^2$?

Consider the following two utility functions: $EU(p)=\sum_i u_ip_i$ $EU^2(p)=(\sum_i u_ip_i)^2$. In preference theory, $EU$ and $EU^2$ are equivalent because they represent the same preference. A ...
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Is Epstein-Zin utility a generalization of dynamic expected utility (DEU)?

Epstein-Zin (EZ) utility is the solution to: DEU is relatively simple: $\sum_t \delta ^t\mathbb E[u(c_t)]$. Is DEU a special case of EZ? How are those two models compared? Since EZ is a solution of a ...
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Most utility functions under risk and uncertainty generalizes expected utility. What is deadly wrong if a model does not include EU as special case?

Why do people generalize EU instead of making an entirely new model, or create a model that is neither a special case nor an extension of EU? To my knowledge, most utility functions under risk and ...
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Auction Theory and Elections

Can we (is it reasonable to) apply auction theory and the various incentive compatible mechanisms to elections and guarantee cardinal voting? I am thinking specifically of VCG auctions and ...
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Risk Aversion under Worst Case Utility Representation

The preference relations (≿A and ≿B) over lotteries is defined as: p ≿ q iff min{v(z) : p(z) > 0} ≥ min{v(z) : q(z) > 0} Under what conditions can you say that ≿A is more risk averse than ≿B?
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How to determine if people behave optimally with a generic utility function?

I have a some real-world data and a real-world choice for which I know the optimum solution is something like $y^* = u(x)$, where $u(.)$ is some utility function. I have data on $y$ and $x$, so if I ...
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Do a group of economic agents really act as if they are rational?

When questioning the rational choice hypothesis, I often get responses that are similar to the followings: "Individuals may sometimes make irrational decisions, but a large group of economic ...
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Can any three of the four vNM axioms (of expected utility theory) be satisfied without satisfying the fourth?

Is it true that any three of the four vNM axioms (of expected utility theory) can be satisfied without satisfying the fourth? Any examples which support such claim? Basically I'd like to prove that ...
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Comparing voting methods when there are only two voters

Consider the Schulze, Kemeny-Young, Ranked Pairs and Borda count voting methods. (The last is obviously the odd one out in this list!) Suppose that there are only two voters. Each voter gives a ...
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Analyzing a Gambling Race Paradox

Suppose a number of players are given $100$ points each, and repeatedly engage in a gamble having positive expected value, with the goals of being the first player to reach $100000$ points. Solving ...
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What is the difference between Impression Management and Signaling Theory?

I'm interested in theories on how organisations shape their stakeholders' (especially consumers' and investors') perceptions and decisions. I read about Impression Management and Signaling Theory. ...
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Pure Nash equilibrium in bidding game?

According to the answer key for a problem set, there is no pure strategy Nash equilibrium in the following problem. Yet I can't see why not. Could it be an error in the answer key? Here's the problem: ...
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Axiom of Minimal Liberalism & Sen's Theorem of Paretial Liberal

Suppose that a person believes that all humans are guaranteed a set of rights that cannot be taken from them in any situation or circumstance (for example, the right to marry a person of your choice, ...
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Question about Social Welfare Function and Social Profile

What are the meanings of a social welfare function and social profile? How are they related?
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If I had a new economical theory, how could I share it with academical environments? I call it “algorythmic economy”

If I had a new economical theory, how could I share it with academical environments? I call it algorythmic economy. I have made this same question on Quora. https://www.quora.com/unanswered/If-I-had-a-...
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Definition of strictly convex preference

Let $x,y\in X$. Does strictly convex preference (which implies that the utility is strictly quasiconcave) mean that: $x\succsim y$ implies $\alpha x+(1-\alpha)y\succ y$ for any $\alpha\in (0,1)$?
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Ordinally Separable Utility Representation

Let $X_i$ be a separable, compact, Banach space. Definition: A weak order $\succeq$ on $X=\prod_{i=1}^NX_i$ has an ordinally separable representation if there exists $u_i: X\rightarrow \mathbb{R}$ and ...
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Can I rename a utility function based on its properties?

A big name researcher gives a name to a specific utility function 30 years ago. Now I am writing a paper and feel that a new name might be more suited because of the properties associated with the ...
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Original formulation of the axioms of rationality [duplicate]

Completeness and transitivity are considered to be the two axioms of rationality in case of decisions under certainty. I wanted to know when were these axioms first proposed, by whom and the first ...
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Ranked choice preference with ties - Arrow's Impossibility Theorem

My question relates to my understanding of Arrow's Impossibility Theorem and ranked choice. It seems to me that the requirements on social choice functions are too strict. A social choice function ...
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Identifiability of Non-Parametric Utility Function?

I recently learned that EU characterized by independence and weak ordering is identifiable, but a utility function like: $U(x)=v_1(x)v_2(x)$ is not identifiable. Does it mean that "cardinal ...
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If a rational preference relation over simple lotteries $\succsim$ are convex then they satisfy independence?

Let´s say there is an uncertain situation with $N$ possible consequences $C = \{C_1, . . . C_N\}$. Assume that there is a rational preference relation $\succsim$ over simple lotteries. I know that if ...
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Certainty equivalence when the utility is semi-continuous instead of continuous

Let $U:\mathbb R^2\to\mathbb R$ be a utility function. If $U$ is strictly increasing and continuous, then it is well known that for any $(x_1,x_2)$ there exists a certainty $(c,c)$ such that $$U(x_1,...
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Who were the first economists arguing that utility maximization is the core of rationality and economic behavior?

I am looking for the first economists arguing that maximizing utility function is the iff condition of rational behavior. I've learned that neoclassical economics is founded on this argument. Is this ...
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What is the point of considering only pure strategies in a game? How could you restrict people from thinking about mixed strategy?

In an experimental setting, how could you effectively incentivize the subjects to not to adopt mixed strategy? I would like to re-emphasize that the question in concern is "how to prevent people ...
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Why utility should be bounded (or unbounded)?

For Expected Utility and SEU, people make axioms to ensure that the utility is bounded. However, I personally believe that the utility function must be unbounded, especially if we are considering ...
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What is the observable definition of "preference" by Frisch?

To make things weird, although Frisch was fully aware of the importance of random distribution in economics relations, he never mention the randomness in binary preference relations! How to define ...
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What additional axiom to GARP do we need to generate a differentiable or smooth utility function

After researching for a while, I find this: https://www.jstor.org/stable/1913607?seq=2#metadata_info_tab_contents They come up with an axiom called SSARP that generates a preference with smooth demand ...
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Blackwell order of information structures

Consider a model where a decision maker (DM) has to choose action $y\in \mathcal{Y}$ possibly without being fully aware of the state of the world. The state of the world has support $\mathcal{V}$. ...
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Can mixed strategies actually predict behaviour of rational actors in non-constant sum games?

I understand how the concept of the mixed NE (mathematically) works. But I don’t understand how we can expect players to behave in a way that would arrive at such an equilibrium. Consider the ...
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How is the defintion of the mean preserving spread (MPS) not too general?

The mean preserving spread is defined as follows: Consider two lotteries g and h. Let $x_g$ und $x_h$ denote the corresponding random variables. Then h is a mean preserving spread (MPS) of g, if: $...
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What is the definition of: 'commodity space'?

I have seen the concept of commodity space being used multiple times in economics, in particular within microeconomics, but I could not find a general definition of it. Based on the examples that I ...
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$a\geq 0$, $x\succsim y$ implies $x+a\succsim y+a$ so the preference is linear?

$\succsim$ is a continuous and local non-satiate weak order. $x,y,a$ are vectors in $\mathbb R^n$ We say $a\geq0$ if all directions of the vector $a$ is greater or equal to zero. We want to prove (...
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Continuity of preferences

Let $\succsim$ be a transitive and reflexive relation on a metric space $X$ with closed upper and lower contour sets. If $\succsim$ is not complete, does it hold that: for all converging sequences ...
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Example of information structure in a one-player Bayes Correlated Equilibrium

Model Consider a game where a decision maker (DM) has to choose action $y\in \mathcal{Y}$ possibly without being fully aware of the state of the world. The state of the world has support $\mathcal{...
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Write search model as a signal density inducing posterior belief

Consider a decision maker (DM) who as to choose an action from the finite set $\mathcal{Y}$ with cardinality $L. $ The payoff the he gets depends on the action chosen and the state of the world. The ...
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1 vote
1 answer
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Set of Bayes Correlated Equilibria when complete information is not available

Model Consider a game where a decision maker (DM) has to choose action $y\in \mathcal{Y}$ possibly without being fully aware of the state of the world. The state of the world has support $\mathcal{...
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-1 votes
1 answer
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Minimizing consumption in a single market( Partial Equilibrium)

Let there be a good X where the optimal consumption is 0; i.e the social costs for any unit provided would always be greater than the utility surplus of the market. We know that prohibiting it( ...
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3 votes
2 answers
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Exact definition of one-player Bayesian Correlated Equilibrium

Consider a game where a decision maker (DM) has to choose action $y\in \mathcal{Y}$ possibly without being fully aware of the state of the world $V$. The state of the world has support $\mathcal{V}$. ...
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The assumptions of Rational Expectations Models

What are the assumptions between rational expectations models and how restricted are there for the following results of economic theory? Where can I find them all gathered in some textbook or in the ...
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Risk neutrality in single-agent choice problem under uncertainity

Consider the following static single-agent choice problem under uncertainty. Let $V$ be the state of the world with support $\mathcal{V}$ and probability distribution $P_V\in \Delta(\mathcal{V})$. ...
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Consider the utility function U(x,y) = y√x [closed]

Draw the indifference curve for U= 10, U=15, U=20. My knowledge of algebra has deteriorated over the last few years of being out of school and I am really unsure of how to answer this. The X value ...
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2 answers
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Sen's property $\alpha$ holds when limited attention in choice?

Consider the limited attention choice framework by Matejka and McKay (2015). This framework can give rise to consideration sets, as roughly summarised below. Consideration sets in the limited ...
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1 vote
1 answer
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Rigorously Defining Newcomb's "Paradox"

I was surprised to find that we did not have a question on this Stack Exchange on this particular problem in decision theory. So I have chosen to add a question on Newcomb's Paradox, though I would ...
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Is the set of optimal strategies convex in a single-agent decision choice problem?

EDITED with insights from the comment below. Consider a decision maker who has to choose an action among $\mathcal{Y}\equiv \{1,2,...,L\}$. The payoff from choosing action $y\in \mathcal{Y}$ depends ...
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Existence of optimal strategy in a choice problem with uncertainty and information structure

Consider a decision maker choosing an action, $y$, from the finiteset $\mathcal{Y}$, possibly without having complete information about the state of the world. More precisely, let $V$ be a ...
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1 vote
1 answer
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Boots' Theory by Pratchett

Consider the following quote from the book, $\textit{Men at Arms}$ by Terry Pratchett. “The reason that the rich were so rich, Vimes reasoned, was because they managed to spend less money. ...
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