8

Let's review the definitions of the two concepts. Let $\sigma$ be a profile of strategies and $\mu$ a system of beliefs. A pair $(\sigma,\mu)$ is a weak perfect Bayesian equilibrium (WPBE) if $\sigma$ is sequentially rational given $\mu$, and $\mu$ is derived from $\sigma$ using Bayes rule whenever applicable. and A pair $(\sigma,\mu)...


5

You could represent the game in extensive form like this: The dashed lines enclose player 2's information set. This encompasses all of player 2's nodes because player 2 observes neither nature's nor player 1's choice. Player 1's information sets are the two singleton nodes because player 1 knows which branch nature has chosen.


5

First of all, you can differentiate between static (essentially all players move simultaneously and only once) and dynamic (essentially non-static) games. An extensive-form game is essentially a game tree. This form of presentation makes sense when looking at games where players move sequentially. However, you could also represent a simultaneous-move game ...


5

So $h$ is just some history of the game. Consider the following game, where Player 1 first decides Heads or Tails, then depending on his choice, a coin is flipped whose outcome and probabilities depends on the choice of Player 1. An example of a history $h$ is $h = Heads$, i.e. after heads is chosen. Consider the very beginning of the game, before Player 1 ...


4

Here is the maximization problem solved by the company : \begin{eqnarray*} \max_{x_h, p_h, x_l, p_l} & p_h + p_l - x_h - x_l \\ \text{s.t} & \theta_hv(x_h) - p_h \geq 0 \tag{IR$_h$} \\ & \theta_lv(x_l) - p_l \geq 0 \tag{IR$_l$} \\ & \theta_hv(x_h) - p_h \geq \theta_hv(x_l) - p_l \tag{IC$_h$} \\ & \theta_lv(x_l) - p_l \geq \theta_lv(...


4

The game is indeed an $8\times 4$-game. A strategy formally specifies what to do at each information set, including those information sets that can never occur under the strategy. In that sense, a strategy is not simply a "plan of action." This point has been famously made in (see Section 2): Rubinstein, Ariel. "Comments on the interpretation ...


3

It is a convoluted definition because condition 3: "At all $h$, if there exists a previously unclinchable payoff x that becomes impossible for agent $i_h$ at $h$, then $C_i^\subset (h) \subseteq C_i^h (h)$." means that at every history if there is a payoff that player $i_h$ could not have secured (or clinched), but was feasible at every previous history ...


3

I think the question has a typo. Whoever wrote it was trying to make you think about the fact that a, c and d are not even properly defined strategies for player B. This is because a strategy must specify what player B would do in all possibly scenarios he might face. However, the typo comes from the fact that b is not part of a NE. If the first payoff ...


3

You've established that a player getting to 8 or 19 seals a win. You want to understand whether player 1 has any options other than starting with 8, to win. Consider it from player 2's perspective. What is player 2's best strategy if player 1 picks a number less than 8? What is player 2's best strategy if player 1 picks 9 or 10? The answer to both, is ...


3

Because you are looking for subgame-perfect equilibria, it is the correct approach to solve this game backwards. There are 2 proper subgames, and you identified the correct Nash equilibria there. However, note that you specified the payoffs, not the strategies. The correct formulation would be $(a_1,b_1)$ instead of $(2,4)$. Next, you substitute the subgame ...


2

There is no possibility to model a "single" player [that] can simultaneously choose different options. Neither in the normal nor in the extensive game form. Different types of a player can be only be used if they are mutually exclusive. So suppose the citizens (one player!) can either be poor or rich and this may affect their pay-offs and thus actions. ...


2

Perhaps I misunderstand something, maybe you do not allow mixed equilibria. That may be strange in games of imperfect information. Consider an asymmetric matching pennies game. Both players show either Heads or Tails. I forgot how to type game matrices in mathjax, but the payoffs look something like this: $\begin{bmatrix} -1,1 & 3,-3 \\ 1,-1 & -3,3 ...


2

Would this (and why or why not) provide an answer to your question?


2

Let the set of players be $N=\{1,\dots,n\}$. According to my understand your description of the game, I take the following statements to be true: Each player chooses a number of players to integrate, not players with specific identities. As a result, the player making the choice may or may not be included in the coalition. Each player has to choose at ...


2

A "part" of an extensive form game that is not a proper subgame because it does not start at a single node but an entire information set would be called "continuation game". This terminology is fairly standard (Perfect Bayesian Equilibrium). However, I think what you are after is a stochastic game which consists of several states. Each state corresponds to ...


1

Perfect recall means that every player remembers her own history.


1

There are a couple of imprecisions. First, there are 3 subgames: two that start with player 2 moving, and the complete game is also a subgame. Second I think that A is a best response for player 1 iff $r\geq \frac{z-y}{x-y}$ otherwise, B is the best response (perhaps this was just a typo). Third, note that the fraction is guaranteed to be positive and ...


1

Imagine you're Player 2. What different "scenarios" could you find yourself in? If Player 1 chooses Right, then Player 2 could observe if Nature played Up or Down. So you have 2 information sets there. If Player 1 chooses Left, then Player 2 couldn't differentiate between (Up, Left) and (Down, Left). How many information sets should there be here? So in ...


1

Let $c = 0$ and $Q_k = \sum_{i = 1}^k{q_i}$. For $j = n$ the best response is given by \begin{align} b_n(Q_{n-1}) = \arg\max_{q_n}(1 - Q_{n-1} - q_n)q_n = \frac{1- Q_{n-1}}{2}. \end{align} For $j = n-1$ the best response is given by \begin{align} b_{n-1}(Q_{n-2}) =& \arg\max_{q_{n-1}}(1 - Q_{n-2} - q_{n-1} - b_n(Q_{n-1}))q_{n-1} = \frac{1- Q_{n-2}}{2}. \...


1

I am adding a new answer to make a different point. I am still not sure what exactly you want to do. You have not formally accepted one of the answers, so here I go. Let me take your example of a government setting a tax. First the government sets a tax $t$. Then, there is a continuum of citizens of measure 1. Each citizen has a type $\theta$ and these ...


1

There are several examples in the literature. In the extensive game form, you can express that two different players move simultaneously by letting them move sequentially, but the later players do not observe what the previous player did. Strategically, it is equivalent whether the player move at the same time or whether one player moves first and the other ...


1

Maybe I'm missing some information about this model, but the intuitive interpretation to me is that $$u_B = v(s) - p $$ and $$u_S = p - c(s)$$ where $p$ is the price paid by the buyer for the good. Pareto optimality is obvious, and it's easy to check that the Pareto optimal quality is positive.


1

I believe you are correct -- though I will say appealing to the one-shot deviation principle here seems a little overpowered. There are only three stages to this game so checking for all equilibria (and not just your conjectured one) in each proper subgame is doable. You do seem to be applying it correctly, however!


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