17 votes
Accepted

Has the Nash Equilibrium led to any significant economic discoveries?

Two areas that have been profoundly affected by game theoretic research stemming from Nash's contribution are Oligopoly theory There are actually a few examples of what would come to be known as ...
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12 votes

Has the Nash Equilibrium led to any significant economic discoveries?

You're not alone in your skepticism of the relevance of game theory. Some of the greats, including Gary Becker, were at times dismissive of the practical/empirical importance of game theory (see the ...
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10 votes

Has the Nash Equilibrium led to any significant economic discoveries?

This is only half a joke : Nash-equilibrium gives a very good prediction on the relative size of groups of foraging ducks on a pond when two food sources are established at opposite sides of the pond....
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9 votes

Difference between Sequential and Weak Sequential (Weak Perfect Bayesian) Equilibria?

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 ...
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9 votes
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Relaxing the notion of Nash Equilibrium

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_{...
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9 votes
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Correlated equilibrium intution

Correlated equilibrium concepts are often related to communication, most easily implemented via a mediator. Communication can archive two things. It helps with coordination and it can transmit private ...
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9 votes

Correlated equilibrium intution

An example to highlight the difference between correlated and uncorrelated random strategies in games of perfect information. Consider a game of Chicken with the payoff matrix Straight Chicken ...
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8 votes
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Is there always a pure Nash equilibrium in a resource selection game?

Yes, there is always a pure Nash equilibrium. See: I Milchtaich (1996). Congestion games with player-specific payoff functions. Games and economic behavior 13 (1), 111-124. You are interested in ...
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8 votes

Is a mixed strategy ever the best response to a pure strategy?

Fixing the strategy of the opponent, a mixed strategy never yields a strictly higher utility if you are expected utility-maximizing. The reason is that the expected utility from a mixed strategy is at ...
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8 votes
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Prove that for every Nash equilibrium $\sigma^*$, the probability distribution $p_{\sigma^*}$ is a correlated equilibrium

A strategy profile $\sigma^*=(\sigma_i^*,\sigma_{-i}^*)$ is a Nash equilibrium if for all player $i$, \begin{equation} u_i(s_i,\sigma_{-i}^*)\ge u_i(s_i',\sigma_{-i}^*), \quad \forall s_i\in\mathrm{...
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7 votes
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Monopolistic and Bertrand (Nash) Competition

The following paper compares the efficiency of the Bertrand and Cournot game in the case of product differentiation. However, their utility function is more general than the Dixit/Stiglitz case. You ...
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7 votes

In a game with alternating moves and complete information, the Nash equilibrium cannot be a non-trivial mixed equilibrium?

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 (...
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  • 4,589
7 votes
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SPNE of a normal form game

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 ...
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7 votes
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symmetry of equilibria with heterogeneous players

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 ...
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7 votes
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Effect of bounding action space on the set of equilibria

Let $N=2$ and for $(x,y)$ and $(p,q)$ in $[0,1]^2$ let $d_{p,q}(x,y)$ be the Euclidean distance between $(p,q)$ and $(x,y)$, i.e. $d_{p,q}(x,y)=[(p-x)^2+(q-y)^2]^{1/2}$. Choose $k>0$ such that $k&...
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  • 4,589
6 votes

Comparing Nash equilibria

Yes there absolutely is. It is generally agreed upon that utility is (at least) ordinal. That means I can compare utility levels for a single person (not necessarily across people) and the numbers ...
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6 votes
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Rosen's Diagonal Strict Concavity condition

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 ...
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6 votes

Rosen's Diagonal Strict Concavity condition

So you want to find a maximum of $\sigma(s,z)$. If $\sigma$ is diagonally strictly concave you can do so by starting at any point and just following the gradient $g(s,z)$ until you find the maximum ...
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6 votes

Is a Nash equilibrium anything more than what it is?

The conventional definition of Bayesian Nash equilibrium (BNE) is as follows: A pure strategy BNE is a profile of type-contingent strategies $$(s_i(\theta_i),s_{-i}(\theta_{-i}))=(s_1(\theta_1),\...
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6 votes
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What is the definition of a "Stackelberg leader-leader equilibrium"?

A leader-leader Stackelberg is a situation in the Stackelberg model where both firms believe they are leaders. This leads to global production being much higher than expected by both firms, as they ...
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  • 1,246
6 votes

Can a game with a unique pure strategy Nash equilibrium also have a mixed strategy equilibria?

...
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  • 5,257
6 votes

Game theory software

I have used gambit (python) in the past and could recommend it. It also includes a small GUI which makes things very intuitive. R also has GameTheory if you prefer this platform instead
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6 votes
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Bayesian-Nash equilibrium in a first-price auction

It is actually assumed that $b_i(v_i)$ is of the form $\alpha_i+\beta_i \cdot v_i$. So it is an affine function. Linearity only works if the bottom of the uniform distribution is 0. A somewhat ...
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6 votes
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Example of a game with no Nash equilibria but at least one correlated equilibrium

There are three players, 1,2,3, and the action spaces of players 1 and 2 are both $[0,1]\times\mathbb{N}$, the action space of player 3 is $[0,1]$. The generic action of player 1 is written as $(x,m)$,...
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6 votes

Interpretation of Solution Concepts

One reason why there exists no ultimate overview is that these issues are still under debate. A great entry point would be the survey "Foundations of Strategic Equilibrium" by John Hillas and Elon ...
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6 votes
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Existence of Symmetric Pure Strategy Equilibrium

There is no guarantee that symmetric games have symmetric equilibria. See this paper for concrete examples. There is also no guarantee that symmetric games have pure-strategy equilibria. For example, ...
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6 votes
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In a game with alternating moves and complete information, the Nash equilibrium cannot be a non-trivial mixed equilibrium?

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 ...
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6 votes
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Finitely repeated Prisoner’s Dilemma with switching cost

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". ...
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6 votes

Is there really a Nash equilibrium in this example?

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....
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