# 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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### About The Bayesian Conditional-Probability Systems in Myerson's Game Theory: Analysis of Conflict

I am self-studying game theory using Myerson's Game Theory: Analysis of Conflict. I got some trouble understanding his Bayesian conditional-probability system. The Bayesian conditional-probability ...
35 views

### In Debreu's representation theorem of ordinal utility, is the assumption of "second countability" necessary?

Debreu 1959 states that: second countability, continuity, and weak ordering sufficiently implies the existence of real (continuous) utility function. The second and third assumptions are also ...
1 vote
44 views

### Proving duality of UMP and EMP arguing with continuity of utility

In Mas-Colell et al.'s Microeconomic Theory Proposition 3.E.1(ii) (p. 58) states that if $\succsim$ is a rational (i.e. complete and transitive), continuous, and locally nonsatiated preference ...
116 views

### Minimal assumption for a “certainty equivalence” exists

Let $R$ be the set of real number. Let $N$ be an infinite set. Let utility $u:R^N\to R$. The utility function is strictly monotonic. My question is, does the certainty equivalence $CE$ exist? Do we ...
1 vote
69 views

### About Theorem 1.1 in Game Theory: Analysis of Conflict by Roger Myerson

I am self-studying game theory using Myerson's Game Theory: Analysis of Conflict. I got some trouble understanding his proof of Theorem 1.1, the Expected-Utility Maximization Theorem. The Theorem goes ...
96 views

119 views

### Information structure for complete information

Model A decision maker (DM) has to choose action $y\in \mathcal{Y}$ possibly without being fully aware of the state of the world. $\mathcal{Y}$ is a finite set. The state of the world is a random ...
1 vote
65 views

### Representation theorem for $\succsim\supset>\cup\sim$

On $\mathbb R^2$, define $x=(x_1,x_2)>(y_1,y_2)=y$ if $x_i\geq y_i$ for all $i$ and $x_j>y_j$ for some $j$. Let $\sim$ be an equivalence relation that $x\sim y$ implies $x\not> y$. Define ...
1 vote
48 views

### Say a preference is "constant“, by analogy with a constant function

Let $f$ be a function with range of $\{-1,1\}$ and $f(x,y)=-f(y,x)$. Let the preference $\succ\subset X\times X$ where $x\succ y \iff f(x,y)=1$. In math we can define $f$ to be a constant function on ...
1 vote
69 views

### 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 ...
259 views

### 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 ...
85 views

### 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 ...
131 views

### 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 ...
103 views

### 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 ...
56 views

### 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 ...
23 views

### 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?
108 views

### 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 ...
239 views

### 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 ...
433 views

### 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 ...
44 views

### 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 ...
83 views

### 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 ...
248 views

### 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. ...
48 views

### 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: ...
1 vote
97 views

### 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, ...
276 views

### Question about Social Welfare Function and Social Profile

What are the meanings of a social welfare function and social profile? How are they related?
45 views

### 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-...
1 vote
81 views

### 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 $\alpha\in (0,1)$?
165 views

### 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 $u_i: X_i\rightarrow \mathbb{R}$ ...
1 vote
82 views

### 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 ...
21 views

### 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 ...
56 views

### 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 ...
1 vote
39 views

### 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 ...
113 views

### 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 ...
133 views

### 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,...
85 views

### 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 ...
335 views

### 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 ...
1k views

### 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 ...
101 views

### 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 ...
1 vote
100 views

### 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 ...
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 $\mathcal{V}$. ...