9

Not really. There are many compact metrizable topologies you can put on this space, but none that relate meaningfully to the structure of the problem. Let's look first at the case $A_1=[0,1]$ and $A_2=\{0,1\}$. Consider the elevation function $e:A_1\times\Sigma\to\Delta(A_2)=[0,1]$ given by $e(a,\eta)=\eta(a)$. If you want the ultimate action choice of ...


8

If in equilibrium, a player "chooses a mixed strategy" that plays $H$ and $T$ with positive probability, $H$, and $T$ must be both optimal choices. It is a standard result that for a (subjective or objective) expected utility maximizer, randomizing can only be optimal if it is over pure optimal choices. This is a direct consequence of expected ...


8

No, there is not. Consider a game with two players, Ann and Bob. Both choose such vectors with entries $0$ or $1$ of the form $(a_1,a_2,\ldots,a_J)$ or $(b_1,b_2,\ldots,b_J)$, respectively. If $\sum_{i=1}^J a_i+b_i$ is odd, Ann wins and Bob loses. If the number is even, Ann loses and Bob wins. Clearly, one of them loses in every profile of pure strategies, ...


8

Your proposed notion of what a strategy should be in repeated game can be adapted to all extensive form games has been called a plan of action by Ariel Rubinstein, who discusses this and related issues in the paper Rubinstein, Ariel. "Comments on the interpretation of game theory." Econometrica: Journal of the Econometric Society (1991): 909-924. ...


7

If you draw the corresponding game tree, you will see that "equivalent to simultaneous move game" implies that the game has no proper subgame and the only subgame is the whole game. This is because the information set of the second player covers every move of the first player. Therefore, every Nash equilibrium is trivially also subgame perfect.


7

the equilibria of the game in which the strategies of users who face the same reward and costs (i.e. same type) are the same. This sounds like an ex ante (or ex interim or ex post) symmetric equilibrium, depending on the timing of the realization of the types. If both dimensions of the types are realized at the beginning of the game, then I'd go with the ...


6

There are a lot of game theorists quite open to many versions of behavioral economics. That being said, I think there are some reasons why these are still fairly separate areas. I will focus in particular on the issue of rationality. Parts of behavioral economics simply do standard economics with somewhat different preferences, such as other-regarding ...


6

Disclaimer: My academic coming of age was in an environment where behavioral economics played only a minor role. My research is theoretical, both "behavioral and non-behavioral", economics. I believe it is incorrect to say that game theorists (or economists) in general are not convinced by behavioral economics. You can easily see it is considered ...


6

The diagonally strict concavity property is better known as the strict monotonicity property of the pseudo-gradient. An operator $\Psi:\mathbb{R}^n \to \mathbb{R}^n$ is strictly monotone if the following holds true: $$\forall x_0, x_1 \in \mathbb{R}^n: \left< x_0 - x_1, \Psi(x_0) - \Psi(x_1) \right> >0. $$ In your case $\Psi$ would be $g$. The ...


6

This statement is wrong. Consider Alternating Matching Pennies with imperfect information (the follower doesn't observe the leader's move). The strategic form of this game is just the classical (simultaneous-move) Matching Pennies Game and the unique NE has both players mixing.


6

It's well known that if $\succsim$ satisfies independence, then it is also convex. Since $\succsim$ satisfies independence, $L\succsim L^{'} \iff \alpha L+(1-\alpha)L^{''}\succsim \alpha L^{'}+(1-\alpha)L^{''}$ for all $\alpha \in [0,1]$ and $ L, L^{'}, L^{''}\in \mathfrak{L} $ Convexity requires: $L\succsim L^{''}$ and $L^{'}\succsim L^{''} \...


6

A couple hints. Regarding the lower bound on $\epsilon$: What happens if deviation occurs at stage $T$? In other words, there is no opportunity for your so-called "punishment stages". Regarding the upper bound on $\epsilon$: Suppose player 2 deviates at stage $T-1$ but player 1 does not. What must be true about $\epsilon$ in order for player 1 to ...


6

The Nash equilibrium is indeed (down, right). Note that your chart has helpfully underlined the max value among all possible strategies each player can play, conditional on what the other player plays. The steps on the graph are useful as well: If no numbers are underlined in a cell, that strategy set is strictly dominated by something else. If one number ...


6

You are only required to get Nash equilibria and not sequentially rational/subgame perfect equilibria. Hence Player 2's actions at information sets that do not occur (that do not reflect Player 1's actual strategy) do not need to be best responses. All you have to make sure is that no one is better off by deviating. In case 4., regardless of what Player 2 ...


6

Let's first determine the sets of actions of the players. An action of player 1 is simply a bid $x_1 \in \mathbb{R}_+$. An action of player 2 is a function: $f_2: \mathbb{R}_+ \to \mathbb{R}_+$ that determines for every action $x_1$ of Player $1$ an action $x_2 = f_2(x_1) \in \mathbb{R}_+$. Let us denote by $F_2$ the set of all actions of player 2. Let's now ...


6

I have never done such purification exercises, but I would approach it like that. As you state, $p=Pr(\epsilon_1>\frac{1-2q}{x})=1-Pr(\epsilon_1<\frac{1-2q}{x})=1-F (\frac{1-2q}{x}),$ $q=Pr(\epsilon_2>\frac{2p-1}{x})=1-Pr(\epsilon_2<\frac{2p-1}{x})=1-G (\frac{2p-1}{x}),$ where $F$ and $G$ are the cdfs corresponding to densities $f=F'$ and $g=G'$. ...


5

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 ...


5

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. ...


5

Level-k reasoning in the stag hunt game is analyzed in Gracia-Lázaro, Carlos, Luis Mario Floría, and Yamir Moreno. "Cognitive hierarchy theory and two-person games." Games 8.1 (2017): 1. The idea that playing $s$ guarantees its payoff is discussed in Aumann, Robert "Nash equilibria are not self-enforcing, in ‘‘Economic Decision-Making: Games, ...


5

$U^S(\bar y,\bar m,b)=\max_{y\in Y}U^S(y,\bar m,b)$ is standard notation that says $\bar y$ is the action (taken by receiver) that would maximize sender's utility given message $\bar m$ and bias parameter $b$. In other words, sender would prefer receiver to choose $\bar y$ when he sends message $\bar m$. Sender's maximized utility is $U^S(\bar y,\bar m,b)$. ...


5

As is clear from the answer of VARulle, complete information is of no use. Every (finite) game in normal-form is the normal form of an extensive form game of complete information. The situation is different for games of perfect information, and one can prove a result to the effect that "Almost all finite games of perfect information have equilibria that ...


5

The real-life question is "how do you persuade people to use mixed strategies"? To stick with your example, Consider a person that has to make a binary choice $(H, T)$, and, after contemplation, they conclude that the optimal strategy is the mixed strategy $(2/3, 1/3)$. I have never know of anyone putting two red and one blue ball in a vase and ...


5

Since you mention IEWDS, I presume by "dominant" you actually mean "dominated". Any strictly dominated strategy would satisfy the condition defining weakly dominated strategies and hence be called such. And yes, strictly dominated strategies can (and should) be eliminated in the process of IEWDS. The possible typo notwithstanding, any ...


5

There is also no NE which sustains coopration for more or less the same reason as in the SPNE case. Consider, a PD played twice. A strategy contains five actions, one for each decision node: one in the beginning (empty history) and one for each of the four period-2 histories (CC,CD,DC,DD). I claim that any strategy other than (D;D;D;D;D) is dominated. ...


5

tldr: The core is a very general concept that can be used in a vast amount of models. Applying it to the setting of a general equilibrium, you can show that every competitive equilibrium is in the core. The core The core is a concept that can be defined for very abstract environments. Consider a population of agents $N$ and an space $\Omega$ of outcomes. ...


5

When integrals look different than what pops into your head, often the reason is integration by parts. For your example note that $$\int_R^1 (\theta -R) g(\theta) d \theta + \int_R^1 G(\theta) d \theta = (1-R) - 0,$$ where the right-hand side is equivalent to $\int^1_R 1 d\theta$. Hence, the two expressions you consider are equivalent. It's of the form $$\...


5

I would say applications of RL in economics are still in their infancy, though probably have applications in many state-based models, agent-based models, or fields with repeated interactions like game theory. There has been some work done on what you might call the "microfoundations" part - which is studying "regret". Susan Athey, for ...


4

As Michael Greinecker noted, the stag hunt is the leading example of a symmetric 2x2-game with a payoff-dominated but risk-dominant NE. In symmetric 2x2 coordination games, a pure NE is risk dominant iff it is the unique best reply to the mixture $(\frac12,\frac12)$. Since Level-0 types are usually assumed to mix uniformly over pure strategies, all higher-...


4

I know a bunch of people who teach behavioral economics using Erik Angner's book. I love Rani Spiegler's book on behavioral economics in IO called "Bounded Rationality and Industrial Organization." Both books will introduce models that can be applied to poker. Perhaps a little lighter but still scientific readings would be Kahneman's "Thinking,...


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