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.
146
questions
1
vote
1
answer
37
views
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 ...
- 191
3
votes
0
answers
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 ...
- 1,796
1
vote
1
answer
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 ...
- 821
2
votes
2
answers
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 ...
- 1,796
1
vote
1
answer
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 ...
- 191
3
votes
0
answers
96
views
Generalization of Debreu's additive utility function $\sum_nu_n(x_n)$ with infinite number of commodities
I want to generalize: $\sum_nu_n(x_n)$.
Here $x_1,x_2,..,x_n,...$ are commodities. There are infinite number of commodities: $n\in\mathbb N$ or $n\in \mathbb R_+$
The following not a candidate: $\...
- 1,796
0
votes
1
answer
22
views
Is the relation $\mathcal{R}=\{(1,2),(2,3),(1,3)\}$ on $X=\{1,2,3\}$ complete?
Is the relation $\mathcal{R}=\{(1,2),(2,3),(1,3)\}$ on $X=\{1,2,3\}$ complete?
By looking at the completeness definition in preference: Definition 1.1(c), this is same as the connected relation in the ...
- 143
0
votes
0
answers
24
views
Can a business's search for profits be considered as movement in state space?
When businesses make decisions to increase profits, they have to adjust several "parameters" of the business such as quantity to output, pricing, choosing a production mix, brand positioning,...
- 391
0
votes
1
answer
34
views
What does the Arrow-Pratt risk aversion measure means in the deterministic case?
What type of preferences that are not related to risk aversion can the Arrow-Pratt measure of absolute (or relative) risk aversion model?
So far, it seems to me that low RRA/ARA preferences imply that ...
- 83
2
votes
0
answers
95
views
Proof of First-Order Stochastic Dominance with Riemann Sums
Let $A = \mathbb{R}$. For $p,q\in \mathcal{L}(A)$, $p$ first-order stochastically dominates (FOSD) $q$ if $F_p(a)-F_q(a) \leq 0, \forall a\in A$.
Show that $p$ first-order stochastically dominates $q$ ...
2
votes
1
answer
72
views
Choice theory vs decision theory
I always thought that Decision Theory and Choice Theory are the same fields. But when reading the Wikipedia entry for Decision Theory recently, I read the explicit clarification: "not to be ...
- 1,801
4
votes
1
answer
71
views
What is the economic intuition of prudence in the static case?
How can we interpret a "prudent" agent in the static case (i.e., someone with $u'''(\cdot)>0$)?
I understand that in a dynamic setting, someone exhibiting prudence would do precautionary ...
- 83
1
vote
1
answer
38
views
What is the risk aversion domain and how this could change in a dynamic market game?
Most of the market microstructure theory models assume a risk aversion coefficient, say $\gamma$ that is indexed with $i$ since any individual $i$ has her own $\gamma_i$ coefficient. Also, the inverse ...
- 182
0
votes
0
answers
30
views
separability question
Suppose an agent’s basic desires pertain to 10 logically independent propositions 𝐴1, 𝐴2, ... , 𝐴10. There are (2 to the power 10 = )1024 conjunctions of these propositions and their negations (...
- 21
2
votes
1
answer
51
views
Lower bound for the utility in a decision problem with uncertainty
Model
Consider a single-agent decision problem with uncertainty.
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}...
- 426
4
votes
2
answers
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 ...
- 426
1
vote
0
answers
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 ...
- 205
1
vote
0
answers
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 ...
- 1,796
1
vote
0
answers
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 ...
- 1,796
4
votes
1
answer
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 ...
- 1,796
3
votes
0
answers
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 ...
- 205
2
votes
1
answer
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 ...
- 1,796
4
votes
1
answer
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 ...
- 1,796
0
votes
0
answers
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 ...
0
votes
0
answers
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?
- 1
3
votes
1
answer
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 ...
- 76
10
votes
3
answers
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 ...
- 460
2
votes
1
answer
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 ...
- 21
5
votes
0
answers
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 ...
- 232
2
votes
1
answer
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 ...
- 21
4
votes
1
answer
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. ...
- 143
0
votes
0
answers
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:
...
- 43
1
vote
2
answers
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, ...
- 41
3
votes
1
answer
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?
- 41
0
votes
2
answers
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
0
answers
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 $\alpha\in (0,1)$?
- 1,796
8
votes
2
answers
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 $u_i: X_i\rightarrow \mathbb{R}$ ...
- 183
1
vote
1
answer
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 ...
- 1,796
0
votes
0
answers
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 ...
- 1,801
2
votes
1
answer
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
0
answers
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 ...
- 1,796
2
votes
1
answer
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 ...
- 35
5
votes
0
answers
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,...
- 1,796
4
votes
0
answers
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 ...
- 1,796
6
votes
3
answers
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 ...
- 1,796
3
votes
1
answer
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 ...
- 1,796
3
votes
0
answers
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 ...
- 1,796
1
vote
1
answer
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 ...
- 1,796
4
votes
1
answer
215
views
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 $\mathcal{V}$.
...
- 426
2
votes
2
answers
87
views
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 ...
- 321