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The Key to BNE is that players that know something (about the state of the world or their type) can condition their strategies with their information. That is, for example, in question 3, type A could choose strategy U, while type B could choose strategy D. Therefore from the second player's perspective, there are four possible pure strategies of his ...

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If you use LaTeX, you can also draw game trees with the istgame package, which is based on TikZ. The manual contains lots of examples with full codes including: game trees in any direction: downwards, upwards, eastwards, -45 degree, etc. labelling players, action labels, and payoffs decision nodes, chance nodes, terminal nodes various information sets ...

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It is true that when both principal and agent are risk neutral, the first best can be obtained despite asymmetric information. You should refer to a textbook, such as MWG (ch.14), for the technical details of such models. I'll give an intuitive explanation here. The intuition of the result lies in optimal risk sharing between the principal and the agent. ...

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The center allocation $z_{i}:=(v(N)/|N|)$ for all $i \in N$ is in the core of the symmetric game $v$, whenever $z(S) = \sum_{i \in S} z_{i} \ge v(S)$ holds for all $S \subseteq N$. Now, if $v(N)/|N| \ge v(S)/|S|$ for all $S \subseteq N$, then it holds $z(S) = \sum_{i \in S} z_{i} = |S|\cdot v(N)/|N| \ge v(S)$. Thus, $\mathbf{z} \in C(v)$. We observe that ...

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Your conjecture seems to be contradicted, at least for small values of $\sigma$. You can draw the function with the following R-code: qq_f = function(x,k,h,sig){ -pnorm(-k, sd=sig)*( (dnorm(h*(1-x), sd=sig))^2 ) - 0.5*dnorm(-k^2, sd=sig)*( 2*pnorm(h*(1-x), sd=sig) -1 )^2 } curve(qq_f(x,k=0,h=1,sig=0.5),col='blue',xlim=c(-1,3),type='l',main="A ...

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The concept of the BCE from their 2016 paper is similar to what you have. I think Bergemann and Morris' intuitive explanation is valuable so I'll paraphrase it here. Each player in the game has a decision rule that chooses an action, $y$, dependent on the state of the world $V$, and the player's information set, which we'll call $S$. This information set ...

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In more detail, payoffs in the table: the first number is associated with the payoff for the first player, the second number on contrary - with the second player's payoff. Assume the second player plays (K,K), then which one you are going to play? L or R? Of course - L (as $3>2$). Then assume the second player plays (K,U), what is the first player's ...

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\begin{array}{|c|c|c|} \hline &L&R\\\hline T&1,1&0,0\\\hline B&0,0&0,0\\\hline \end{array} In the game above, there are two pure strategy Nash equilibria: $(T,L)$ is an equilibrium in weakly dominant strategies; $(B,R)$ is an equilibrium in weakly dominated strategies. Noting that "dominant" and "dominated" are two different words,...

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It depends on what you assume the firms know about each other. Under full information / rationality assumptions, with identical firms, second-mover advantage disappears because the first-mover will recognize the credible threat of undercutting, and set P = MC and that will be that. If the firms are not identical (e.g., different marginal costs) then turn ...

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The current mainstream theory of value is the subjective theory of value: goods or services have value that people subjectively believe that they have. Rembrandt paintings cost millions because someone subjectively thinks they are worth that much. If there are no market failures the market price captures the subjective value through supply demand ...

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This thread has some game trees. In case you want put them on your documents, you can do like this (using LaTeX with istgame package): \documentclass{standalone} \usepackage{istgame} \begin{document} \begin{istgame}[font=\scriptsize] \setistgrowdirection{east} \cntmdistance{20mm}{20mm} \cntmAistb{q_1=0}[at end,below]{q_1=1,000}[at end,above] \...

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In direct mechanism agents directly report their preferences (preferences are observable). In indirect mechanism agents don’t report their preferences directly. Preferences can be observed only indirectly through signals or behavior. By Revelation Principle if some outcomes can be implemented in indirect mechanism they must be also implementable in the ...

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The dimension of the normal form game derived from an extensive form is given by the number of pure strategies each player has. Generally speaking, the number of pure strategies a player has in an extensive form game equals the product of the number of actions at each information set where she moves. Suppose a player moves at $N$ information sets on a game ...

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First, a caveat: I'm on the job market this year in the midst of the couple weeks when calls are rolling in. Hence, this seemed like a nice way to kill some time (semi-)productively. This is also a disclaimer in case I've made an error :) Now, let's look at the one you're suggested to try for, where player $1$ chooses $0$ with $p$ and $1$ with $1-p$. Again ...

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The article Quality of Information and Oligopolistic Price Discrimination by Liu and Serfes covers this topic in great detail. It also has a rather nice literature review.

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Captive insurance is an insurance for purpose of insuring the owners of the said insurance. It’s a kind of self-insurance where multiple agents pool their capital together to insure themselves. So the term in that sentence refers to the pool of resources for insurance. It’s called captive because the modern concept of this kind of insurance was first ...

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From Bolton and Dewatripont Contract Theory (2005, p.135): "In the absence of risk aversion on the part of the agent and no wealth constraints, the first best can be achieved by letting the agent "buy" the output from the principal." This quote is in the context of a simple two outcome model.

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Step 1. Determine strategies. Step 2. Calculate payoffs for strategy profiles. Step 3. Write normal form.

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Just for your convenience If you are interested in the istgame package, you can do like this (to draw Amit's examples): For the centipede game: \documentclass{standalone} \usepackage{istgame} \begin{document} \begin{istgame} %% for arrows (optional) \xtShowArrows \xtShowEndPoints[ellipse node] \xtHideTerminalNodes %% some more optional settings \...

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I have used LaTeX with tikz package. The following code is used to generate this Centepede Game : \documentclass{article} \usepackage{tikz} \usepackage{bodegraph} \usepackage[printwatermark]{xwatermark} \begin{document} \begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=1.3cm, thick,main node/.style={circle,fill=blue!20,draw,...

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What is meant by fixed point reasoning here? Specifically, Nash equilibria. One way to define a Nash equilibrium in words is "a strategy profile from which no player can be made better-off by unilaterally deviating to a different action". In other words, a Nash equilibrium is a "fixed point" because if the game ends up there, play stops because there are ...

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You have specialized the definition of BCE in two dimensions: there is only one player, and the player has no private information. If you want to allow for private information you can let the player have some signal $\pi:\mathcal{V}\rightarrow\Delta(T_i)$ And let the decision rule $P_{\mathcal{Y},\mathcal{T},\mathcal{V}}\in\Delta(\mathcal{Y}\times \mathcal{... 1 Suppose you are one of the bidders and fix the strategy of the other one to be the one supposed. Your payoff will be$\mbox{Pr(winning the auction, i.e., b>b')}(v-b)$, with b being your bid and b' being the other's bid and v being your valuation (or type). Now, imagine you are the type-5 player. If you bid an integer less than 4, you lose for sure ... 1 As Herr K pointed out your premise is not exact, it is not about GSPs but rather SPs. But even for SPs, I do not understand your examples. What you show is that if the bidders choose the same strategies in example 1 and example 2, the same bidder wins and pays the same amount. This is not surprising: the outcome (who wins and pays how much) depends on the ... 1 The surplus is the value of the object obtained minus the price paid. In your first example, the surplus would be$10-7 = 3$. The reason why you don't find examples where BOTH players are not using optimal strategies is because$b=v\$ is weakly dominant. Unlike a lot of games where you need to consider best response functions, this one can be solved simply by ...

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One way to represent the game tree (assuming that the congressman observes the expert's choice): The game tree is drawn using LaTeX with istgame package. \documentclass{standalone} \usepackage{amsmath} \usepackage{istgame} \begin{document} \NewDocumentCommand\vpay{m} { \begin{matrix} #1 \end{matrix} } \begin{istgame}[scale=2,font=\scriptsize] \...

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