17
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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
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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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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
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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
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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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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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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
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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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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 ...
8
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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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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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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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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
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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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6
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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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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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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
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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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6
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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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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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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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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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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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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
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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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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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