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We have that ${\cal I} = ((X^i)_i, \mu)$ and ${\cal J} = ((Y^i)_i, \nu)$ are two information structures.
An Interpretation mapping for player $i$ is a mapping $\phi^i: X^i \to \Delta(Y^i)$ so it associates with every $x^i$ a distribution over $Y^i$.
Let $x^i \in X^i$. Then $\phi^i(x^i)$ is a distribution over $Y^i$ so $\phi(x^i)(y^i)$ is the ...
4
Suppose you are an analyst studying a Bayesian game. You know the players, the possible states of nature, the common prior, the action spaces, the payoff functions, and you know about some information channels available to the players, the latter given via some information structure. However, you can't rule out that the players have additional information ...
4
One of the most fundamental distinctions in economics is that between stocks, measured at a point in time, and flows, measured over a period of time or as instantaneous rates. The construction of dynamic economic models naturally leads to differential equations in which the rate of change of a stock variable depends on one or more flow variables.
An example ...
4
Well, according to what I see, let us take each part of the value function by its own and then
$$\underbrace{\pi((t_1,t_1)|G)}_{1/2}\underbrace{\sigma((a_1,a_2)|(t_1,t_1),G)}_{3/5\times 3/5=9/25}\underbrace{u_1((a_1,a_2),G)}_{8}=36/25$$
$$\underbrace{\pi((t_1,t_2)|G)}_{1/2}\underbrace{\sigma((a_1,p_2)|(t_1,t_2),G)}_{3/5\times 2/5=6/25}\underbrace{u_1((a_1,...
3
A lot of cooperative game theory deals with transferable utility games, where players can easily transfer "utility" or "payoff" to each other. Another class is non-transferable utility (NTU) games where you get what you get after the strategy profile is implemented and that is that, there are no side transfers.
In NTU games it is sensible ...
2
The simplest example is $-x^3$ in the single variable case, or $-x_1^3-x_2^3$ in the case of two variables. The Hessian matrix is negative semi-definite at $(0,0)$, but there is no maximum at this point.
2
Given that
$T^i$ is $i's$ finite set of messages
and the mechanism uses the signal function to map $T$ to a signal, the architects of the mechanism can use any arbitrary label for the individual messages. In simple game theoretical examples, the strategies of player 1 are often denoted $a_1,a_2...$ This does not tell us anything about what player 1 is ...
2
Hi: The connection between convex sets and convex ( and concave ) functions is that convex ( and concave ) functions should be defined on convex sets because, if the sets are defined on sets that are not convex, then there may be points on the set where the convex (and concave ) function is not defined.
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Yes, compactness is sufficient and no need to restrict yourself to subsets of $\mathbb{R}$.
The proof is by contradiction. Assume that $\succeq$ has no greatest element. Consider the sets:
$$
V_x = \{y \in X| x \succ y\}
$$
These are open sets as $\succeq$ is continuous. If $(V_x)_{x \in X}$ do not cover $X$ then there is an $y$ that is no set $V_x$ for all $...
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