Questions tagged [game-theory]

Game theory is a study of situations of strategic interaction between two or more players in which there is a predefined set of rules and an outcome associated with each choice taken.

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How to find mixed optimal strategies in this zero-sum game?

I'm trying to solve this problem from last year final exam in game theory: Consider the zero-sum game $G=(X, Y, g)$ where $X=Y=[0,1]$, and $$\forall (x,y) \in X \times Y: g(x, y)=\max \{x(1-2 y), y(...
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Who is the first person/paper to introduce “mixed strategy”?

Who is the first person/paper to introduce "mixed strategy"? The PNAS by Nash used this notion without citing anyone. Does the earlier book: Von Neumann, J., and Morgenstern, O., The Theory of Games ...
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Is there some definition about risk sharing?

I was searching for a definition of risk sharing and I have found the following: $\underline{Definition:}$ Risk Sharing — also known as "risk distribution," risk sharing means that the premiums and ...
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What is the implicaiton of agent risk neutrality in moral hazard?

I heard that risk-neutrality of agent in moral hazard imply that the action associated with first-best and the one associated with second-best will be the same. Is it true and why is it so? Are there ...
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27 views

What does “steep incentive contract” mean in the context of adverse selection?

This term was mentioned in a slide and when I looked online, the only thing which I could find talking about this is this paper. For example,in the cost-plus-incentive contract $$x(T, c) = a(T) - a'...
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Finding Bayesian Nash Equilibrium

I'm recently new to Game Theory and I've recently started teaching myself about Bayesian Nash Equilibirum. I've stumbled across a problem set that I can't seem to wrap my head around concerning ...
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What is “indirect mechanism” in the context of adverse selectoin?

On this slide deck, page 12, it says "then this direct mechanism implements exactly the same allocation as the indirect mechanism we started". What is indirect mechanism in this context and in ...
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If the players are symmetric and the core is nonempty, then $x_i = v(N)/n$ for all $i$ is a Core element

Consider a TU game $\langle N,v \rangle$ with $N = \{1,2,\ldots,n\}$ being the set of players and $v : 2^N \to \mathbb R$, $v(\emptyset) = 0$ the characteristic function. The Core of $v$ is defined by ...
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How to write down a normal form of the game with information set?

How can we write down a normal form for this following game with information set? (The picture is modified from a picture from Wikipedia.)
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32 views

Does the case of strategic complements always result in a situation where the Stackelberg leader is worse off?

For example, in the case of Bertrand competition, is the Stackelberg leader worse off relative to the follower? If so, why? Is this because the first mover can always undercut until marginal cost is ...
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Renewal of multiple notions of value in contemporary (green, public good) economy and are there general game theories?

https://en.wikipedia.org/wiki/Exchange_value mentions 4 notions of value: economic value, exchange value, use value, price. But this article also mentions that these are Marxian concepts and the ...
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84 views

Quasi concavity of utility function

I am looking for a method to prove that the following function is quasi-concave in $\alpha$ (or find conditions under which it is true): $ \pi=F(-k)(f(0)^2-f(h(1-\alpha))^2)+ \frac{1}{2}-\frac{1}{2}...
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Is it possible to find a nash equilibrium that is not an equilibrium in weakly dominant strategy?

I know that it is possible to have a Nash equilibrium which is not an equilibrium in dominant strategy, but is it also applicable for equilibrium in weakly dominant strategy (i.e. a Nash equilibrium ...
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1answer
30 views

Difference between equilibrium and k-rationalizability

I'm reading a Structural Models of Nonequilibrium Strategic Thinking: Theory, Evidence, and Applications by Crawford, Costa-Gomes and Iriberri. They write the following: In two-person games, a ...
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Exact definition of one-player Bayesian Correlated Equilibrium

Consider a game where a decision maker (DM) has to choose action $y\in \mathcal{Y}$ possibly without being fully aware of the state of the world $V$. The state of the world has support $\mathcal{V}$. ...
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Why does diamond paradox with two firms have only one subgame?

Diamond paradox as introduced by Diamond in 1971 has only one subgame. Is that because in the first step of the game, the two firms set prices which is unknown for the consumers (who are the players ...
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How to read nash equilibrium from a normal form?

For example, on the section with the title "sequential games in normal form" of this wikipedia page, there is a table with all its SPNE and nash equilibrium labelled. How are the NEs obtained? It ...
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25 views

How to interpret the intersected information partition element of different actors?

In this paper, it said that given $\omega$ is the current state of the world and $P_1(\omega)$ is the unique element from the information partition $\textit{P_1}$ of actor 1 in which the player 1 is ...
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51 views

Questions about the Edgeworth box

I have some doubts about the Edgeworth box, it's a newbie one so thanks in advance for the patience. In case, I'm confused if we can count the root of the contract curve (point M) as one of the ...
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1answer
27 views

First-price symmetric auction with discrete value

I've been trying to work the following problem out but I can't quite seem to understand it, or the whole concept of first-place auctions. I don't understand how we get to the equilibrium. The problem ...
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Strategic Complementarities and Multiplicity

Consider a coordination game, in which actions taken by players are strategic complements. Moreover, assume that actions are strong strategic complements, in that the slope of the best response ...
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Penance strategy game theory

I am having some difficulty in understanding the following use of the penance strategy as a strategy to support compliance in the repeated game presented in Asheim et al. (2006). I understand penance ...
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Is a mixed strategy ever the best response to a pure strategy?

Suppose you are playing a game against an opponent whom you know only uses pure strategies. My question is, is there any such game in which using a mixed strategy in response is better than all the ...
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Cardinal Voting, Incentive Compatibility and Secrecy

Is there any available/feasible/practical way to make a Cardinal Voting both Incentive Compatible and Secret? A method to make a cardinal voting incentive compatible would be to force them to put ...
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What is equilibrium dependent upon in Generalized Second-Price Auctions?

Theory states that GSP auctions induce truthful bidding. Is it the case that this is true ONLY IF a) each of the bidders truthfully bids their value ($b_i = v_i$) (each bidder's optimal strategy) ...
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Expected Utility and Jensen's Inequality

Consider two random variables (costs and valuations) distributed $v\backsim G(.)$ and $c \backsim F(.)$ with pdfs $g(.)$ and $f(.)$. Let the supports of $c$ and $v$ be $[x,y]$. Let $x<a=E(v)<b&...
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Existence of symmetric trembling hand perfect equilibria

Consider symmetric and finite game. By Nash (1950), the game must have at least one symmetric equilibrium (proof). Also, it must have at least one trembling hand perfect equilibrium (proof). ...
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1answer
35 views

Clarification of iterated prisoners dilemma

In the description of an iterated prisoner's dilemma on Wikipedia, it states that in order for a game to be considered iterable, it must conform to the rule 2R > T + S, where R is the reward for ...
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132 views

Evolutionary stable strategies

I am new to evolutionary game theory so I can't figure out whether I'm looking at things correctly. I have the following payoff matrix: $$ A = \begin{matrix} 3 & 0 \\ 5 & 1 \end{matrix} $$ ...
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1answer
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Risk neutrality in single-agent choice problem under uncertainity

Consider the following static single-agent choice problem under uncertainty. Let $V$ be the state of the world with support $\mathcal{V}$ and probability distribution $P_V\in \Delta(\mathcal{V})$. ...
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1answer
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Bayesian Correlated Equilibrium in a one player case: relation with literature

I am curious about the connection between one-player Bayesian Correlated Equilibrium (hereafter, BCE) introduced by Bergemann and Morris for a generic $n$-player setting with $n\geq 1$ (here) and the ...
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Cournot Duopoly and Non existence of best response function

Is it possible for a cournot duopoly game to not have the best response functions? I have a profit function whose first-derivative is piece-wise. When I solve for one player's quantity (decision), it ...
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1answer
50 views

Signaling in zero-sum games?

The foundation of this question is a bit vague (based on something overheard at a conference) but I'm hoping someone here can provide some clarification. I overheard a conversation between two ...
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Who took utimatum game and dictator game as the evidence against Homo Economicus assumption of individual utility maximization?

Wikipedia and this McGill University page states that the two games "have been taken as both evidence for and against the Homo economicus assumptions of rational, utility-maximizing, individual ...
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How does your dominant strategy change in a second price auction if there are two items?

Suppose there are 4 individuals and two items in this particular auction. I understand that the dominant strategy in second price auction is to bid your true value if there is a single item but I'm ...
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Adverse selection problem

The classic literature refers to the problem where information asymmetry exists between an informed and an uninformed counterpart as the adverse selection problem, but how can we verify what kind of ...
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A question about the proof of minmax theorem

I'm reading the lecture note about minmax theorem in zero-sum game. First are the relevant definitions used in my lecture notes: A finite zero-sum game, or matrix game, is given by a real-valued ...
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26 views

Cross derivatives of payoff functions in potential games

In Monderer and Shapley (1996) (pdf) there is the following theorem. The next theorem is well-known (and very useful). THEOREM 4.5 Let $\Gamma$ be a game in which the strategy sets are ...
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Sequential Price Competition for Perfect Complements

There are two goods, $1$ and $2$ produced by two firms at zero marginal costs. The goods are perfect complements. The demand for each goods is: $Q_1=Q_2=a-(p_1+p_2)$. The prices are set sequentially, ...
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Bertrand Duopoly Equilibrium for Discrete Prices

There are two identical firms, $1$ and $2$, with zero marginal costs. They produce homogenous product, which is demanded by a unit mass of identical consumers, each of which has inelastic unit demand ...
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Modeling risk aversion. Expected utility function

I want to model an Expected Utility Function for risk aversion but my problem is uncertainty in itself. I want a function(a special case) $$f(x, y) =\left\{\begin{matrix} h(bx^{1-C}+ay^{1-C}),C\neq 1\...
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1answer
52 views

Framing Effect Risk-Aversion Risk-Pursuit

I am an economics' graduate seeking to study Law and I want to illustrate the importance of legal certainty. Penalties, Costs are negativelly framed. I am trying to word. 200 dollars with 50% ...
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Unique Nash-equilibria in multi-unit auctions with uncertain participation

Setup Consider a one shot sealed bid multi-unit auction where $N$ bidders compete for $K$ identical objects and each bidder $i$ has demand $d_i\in \{1,\dots,K\}$. Bidders receive private i.i.d. ...
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What do game theorists think of behavioural economics?

Though behavioural economics has been popularized over the past decade in both industry and pop culture, it seems that not much has changed with regard to the standard economic definition of ...
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1answer
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Rigorously Defining Newcomb's “Paradox”

I was surprised to find that we did not have a question on this Stack Exchange on this particular problem in decision theory. So I have chosen to add a question on Newcomb's Paradox, though I would ...
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Is Bayes correlated equilibrium relevant for dynamic games?

Consider the following classification of solution concepts ...
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How far must one reason up the belief hierarchy?

In games of imperfect information, one must reason about the belief on the state of nature, the beliefs of other player's beliefs of their beliefs, and so on. This is referred to as the belief ...
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1answer
59 views

Is the set of optimal strategies convex in a single-agent decision choice problem?

EDITED with insights from the comment below. Consider a decision maker who has to choose an action among $\mathcal{Y}\equiv \{1,2,...,L\}$. The payoff from choosing action $y\in \mathcal{Y}$ depends ...
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Trembling hand perfection and weakly dominated strategies

It is well known that players cannot use weakly dominated strategies in a trembling hand perfect equilibrium. My question, however, is a little different: does iterated deletion of weakly dominated ...
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Moral hazard, bubbles, and economies of scale in limited companies and organizations

http://eprints.lse.ac.uk/100058/1/Goodhart_CEPR_DP13494.pdf What would be the disadvantage of limiting the liability to the product of the proportion of the company and the debt(a person with 2% of a ...