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The following two claims hold in the general $n$-shop case. Claim 1. In equilibrium a shop closest to an edge (0 or 1) cannot be alone. Proof. Such a shop could gain customers by moving slightly inward. Claim 2. In equilibrium at most two shops can be in any location. Proof. Assume there is an equilibrium where there are three or more shops in a location. ...


4

Osborne and Rubinstein has a book called "Bargaining and Markets" where they have a detailed exposition on bargaining models. I think its free to download from Ariel's website. Apart from that, Abhinay Muthoo has a very nice book on bargaining: "Bargaining Theory with Applications". You can always look up chapters on bargaining in Tirole'...


4

At the intersection of differential equations and game theory one can find differential games. Arguably, the most famous applications of differential games are in warfare, e.g., the homicidal chauffeur problem. However, not all differential games are of the pursuit-evasion kind. Whoever wants to learn differential games may wish to learn optimal control ...


3

In the second period, the buyer accepts any offer $s_{2}$ $\leq$ $0.7$ if he has rejected the first offer and any offer $s_{2}$ $\leq$ $0.3$ if he has accepted the first offer. Given this, there are only two offers that may be optimal for the second seller: either $s_{2}$ $=$ $0.3$ or $s_{2}$ $=$ $0.7$. Let $\mu$ denote the probability that the second seller ...


3

Suppose player $i$ plays the mixed strategy $\mathbb{P}_i(B)= p_i$, and assume for now that the support of $\mathbb{P}_i$ is $\{B,F\}$ (i.e. player 1 plays a fully mixed strategy). For both $B$ and $F$ to be in 1's support, he must obtain the same expected payoff from either strategy (otherwise, he would put all the weight on the strategy with the higher ...


2

Two points: An Incomplete-Information Game is not well-defined without specifying the beliefs of the players. This is specially relevant if you want to use the standard Bayesian equilibrium techniques. There is still significant leeway here - you may define subjective probabilities for each players, players may have inconsistent beliefs (look at Yildiz's ...


2

The best way to visualise what's going on would be to use Harsanyi's transformation. I'm not drawing the game tree here (but I think Tirole has it in his example). Let's set up notations first. We will denote player 1's strategy by $x=Pr(T)$. We will call the decision of player 2 following the realisation of game $i$ by $y_i=Pr(T)$ - i.e. player 2, following ...


2

Matrix looks correct. To final all pure strategy BNEs, you'll have to discuss cases based on the value of $p$. For example, if $p\in(0,1)$, then $FT$ is player 2's unique best response to $F$. Thus, to have a BNE, you'd want $F$ to be player 1's best response to $FT$ as well, meaning that you'd require $3p>1-p$, or $p>\frac14$. Hence, $(F,FT)$ is a BNE ...


2

Generally, it's a combination of 2 and 3. You can't predict that a stock will go up and down in cycles. Stock prices move randomly. Let's say you're looking at one stock, and it goes down so your algorithm buys some and waits for it to go back up. What happens if it never goes back up higher than it started as? Then you lost money. Even worse: Even if it ...


2

The area of research you are looking for is market microstructure, the study of how markets work and the process of price formation. Naturally, this means studying liquidity, information diffusion, uncertainty, and dynamic games. There is not a lot of mechanism design in market microstructure -- though work along those lines would certainly be welcomed. For ...


2

There is no apriori reason for the Best Response map to be a contraction in general. Here's a simple example (since Battle of Sexes has been my go-to for the past few days): $$ \begin{array}{c|lcr} \text{Player1/Player 2$\rightarrow$} & \text{F} & \text{T} \\ \hline \text{F} & 3,1 & 0,0 \\ \text{T} & 0,0 & 1,3 \\ \end{array} $$ Denote ...


1

To prove that that some given combination of strategies is a Nash equilibrium you don't need to use a fixed-point theorem (such as Brouwer's or the fixed-point theorem for contractions on Banach spaces). What you do have to do, is check that they are best responses to one another. This true for mixed and pure strategies. You also seem to be asking how you ...


1

The two questions you ask are related but distinct. First recall what a separating equilibrium is: its an equilibrium where distinct type takes distinct actions, supported by a system of consistent beliefs. Now lets come to your questions one by one: In the Lemons Market, a separating equilibrium would involve the seller setting different prices for the ...


1

This sort-of happens when a currency is pegged (or similar). The central bank tries to keep its currency within the band, and it is profitable to trade on that basis. So long as investors believe that the band can hold, they will keep the price of the currency within that band on their own. However, if the credibility of the peg is questioned, it can be ...


1

why would a player want to balance out the payoffs of another player I don't think anyone is saying that a player wants to do this. But in mixed equilibrium their strategy is such that this property holds. Without this property, any mixed strategy of the other player would be suboptimal.


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