What is a "solution" for a game? What is an equili
While finalizing a joint paper, a discussion among me and the remaining authors started. Interesting enough to share it with others.
The central issue is the following, already mentioned in the title.
What does it mean a solution for a game?
And when a solution can be termed as an equilibrium?
These questions are quite vague, and they can receive truly distinct answers, depending on how one interprets the questions themselves.
I am open and interested in all possibilities, but I suggest to keep for the moment the discussion as much focused as possible on a specific interpretation, and precisely to discuss at a formal level.
Moreover, as a first start, I would suggest to confine the discussion to the very specific setting of games in strategic form, with a finite number of players and described using payoffs. That is:
$(N, (X_i)_{i \in N}, (f_i)_{i \in N})$
Even better, I would restraint to two-players games $(X,Y,f,g)$, unless it is necessary to use the general case.
Too obvious to say, but of course I will refrain from expressing my viewpoints for some days...
The central issue is the following, already mentioned in the title.
What does it mean a solution for a game?
And when a solution can be termed as an equilibrium?
These questions are quite vague, and they can receive truly distinct answers, depending on how one interprets the questions themselves.
I am open and interested in all possibilities, but I suggest to keep for the moment the discussion as much focused as possible on a specific interpretation, and precisely to discuss at a formal level.
Moreover, as a first start, I would suggest to confine the discussion to the very specific setting of games in strategic form, with a finite number of players and described using payoffs. That is:
$(N, (X_i)_{i \in N}, (f_i)_{i \in N})$
Even better, I would restraint to two-players games $(X,Y,f,g)$, unless it is necessary to use the general case.
Too obvious to say, but of course I will refrain from expressing my viewpoints for some days...
Risposte
Non capisco. Una soluzione non e' un equilibrio del gioco? Oppure ti interessa la differenza tra outcome (ossia cosa viene fuori quando tutti gli agenti giocano strategie di equilibrio) ed equilibrio (ossia strategy profile, beliefs, ecc.).
I'm inclined to consider a solution as a rule that sorts out a set of strategy profiles.
However, my main concern is not on the difference that you pointed at.
An example of a solution for a game that I would not call "equilibrium".
I will consider the universe of games sketched in my opening post, with 2 players only.
A solution for a game $G$ is (Fioravante,Patrone) if (Fioravante,Patrone)$\in X \times Y$. Otherwise, it is the empty set.
Leaving aside the ambiguity in the formulation (am I referring to a string of alphanumeric signs, or to someting else?), I hope that no one would bring this idea of a solution under the umbrella of an "equilibrium" solution.
A less (?) provocative example. A solution for a game $G$ is (any element in) the set of the couples maxminimizing payoffs for the players.
I mean: for player $I$, I look at $\bar x$ s.t. $\min_(y \in Y) f(\bar x, y) \ge \min_(y \in Y) f(x, y)$ for all $x \in X$.
Similarly for player $II$. And a solution will be any couple like that, provided it exixsts (included that the mentioned minima do exist).
Does this correspond to an "equilibrium" idea?
However, my main concern is not on the difference that you pointed at.
An example of a solution for a game that I would not call "equilibrium".
I will consider the universe of games sketched in my opening post, with 2 players only.
A solution for a game $G$ is (Fioravante,Patrone) if (Fioravante,Patrone)$\in X \times Y$. Otherwise, it is the empty set.
Leaving aside the ambiguity in the formulation (am I referring to a string of alphanumeric signs, or to someting else?), I hope that no one would bring this idea of a solution under the umbrella of an "equilibrium" solution.
A less (?) provocative example. A solution for a game $G$ is (any element in) the set of the couples maxminimizing payoffs for the players.
I mean: for player $I$, I look at $\bar x$ s.t. $\min_(y \in Y) f(\bar x, y) \ge \min_(y \in Y) f(x, y)$ for all $x \in X$.
Similarly for player $II$. And a solution will be any couple like that, provided it exixsts (included that the mentioned minima do exist).
Does this correspond to an "equilibrium" idea?
Sono ancora un po' confuso. Secondo le tue definizioni, qualsiasi strategy profile e' una soluzione no? Se non ho capito male, per una soluzione basta una regola che "tiri fuori" uno strategy profile. Pero' se sulla regola non imponi alcuna restrizione (cioe' se qualsiasi regola va bene), allora per ogni strategy profile puoi sempre trovare una regola (anche molto banale) da cui risulta lo strategy profile che volevi.
Non conosco definizioni formali e generali del concetto di equilibrio (e confesso che non mi sono speso molto per cercarle). Credo che l'idea dovrebbe essere di qualcosa che resta "immobile" se "nel sistema" non cambia niente (ma mi rendo conto che detta cosi' e' molto grossolana). Intuitivamente, in un gioco qualsiasi cosa che non e' un equilibrio di Nash non sembrerebbe essere un "equilibrio", perche' ci sarebbe sempre un giocatore che avrebbe un incentivo a effettuare una one shot deviation, quindi il "sistema" non resterebbe "immobile". (Ma non bastonatemi per la paurosa mancanza di rigore ... .)
Comunque, per tornare a bomba, ho l'impressione che il concetto di soluzione non sia molto utile. Mi sembra che basti il concetto di strategy set . Sbaglio?
[PS: ma va bene se scrivo in italiano?]
Non conosco definizioni formali e generali del concetto di equilibrio (e confesso che non mi sono speso molto per cercarle). Credo che l'idea dovrebbe essere di qualcosa che resta "immobile" se "nel sistema" non cambia niente (ma mi rendo conto che detta cosi' e' molto grossolana). Intuitivamente, in un gioco qualsiasi cosa che non e' un equilibrio di Nash non sembrerebbe essere un "equilibrio", perche' ci sarebbe sempre un giocatore che avrebbe un incentivo a effettuare una one shot deviation, quindi il "sistema" non resterebbe "immobile". (Ma non bastonatemi per la paurosa mancanza di rigore ... .)
Comunque, per tornare a bomba, ho l'impressione che il concetto di soluzione non sia molto utile. Mi sembra che basti il concetto di strategy set . Sbaglio?
[PS: ma va bene se scrivo in italiano?]
Of course you can write in Italian!
I do not agree completely on what you say about "my" use of the term solution, but for sure you have got the point: "anything" can be termed as a solution. And the examples that I gave for sure do not correspond to an idea of equilibrium.
I agree with you that a good idea is to refer to the common sense of the term "equilibrium". An idea could be to say that a solution is an equilibrium-like solution if it is a rest point of a conveniently defined (discrete?) dynamical system.
You pointed at one possible dynamical system: the one which underlies the "best reply dynamics". Clearly from this we get the Nash equilibrium.
But I hope you agree with me that it is a too specific dynamic system, taylored for the Nash equlibrium.
So, suggestions about other dynamical systems? About different ideas of equilibrium?
Other approach (the extensive one): which of the solutions that are used for strategic games would deserve the name of "equilibrium"?
I do not agree completely on what you say about "my" use of the term solution, but for sure you have got the point: "anything" can be termed as a solution. And the examples that I gave for sure do not correspond to an idea of equilibrium.
I agree with you that a good idea is to refer to the common sense of the term "equilibrium". An idea could be to say that a solution is an equilibrium-like solution if it is a rest point of a conveniently defined (discrete?) dynamical system.
You pointed at one possible dynamical system: the one which underlies the "best reply dynamics". Clearly from this we get the Nash equilibrium.
But I hope you agree with me that it is a too specific dynamic system, taylored for the Nash equlibrium.
So, suggestions about other dynamical systems? About different ideas of equilibrium?
Other approach (the extensive one): which of the solutions that are used for strategic games would deserve the name of "equilibrium"?
Ho capito quale e' il tuo punto. Sicuramente saprai che la cosiddetta evolutionary game theory si e' posta in parte questo problema affrontando i giochi con modelli esplicitamente dinamici. Per un articolo bello, non piu' recente ma non ancora datatato, che affronta il rapporto tra Nash equilibrium, learning, comportammneto reale degli agenti ecc. ti suggerisco di guardare:
Mailath G., "Do People Play Nash Equilibrium? Lessons From Evolutionary Game Theory", Journal of Economic Literature, 36 (September 1998), 1347-1374
In particolare c'e' una sezione su Nash Equilibrium as an Evolutionary Stable State, che forse si avvicina a cio' a cui stai pensando tu.
Una cosa pero' ancora mi sfugge. Secondo te, il concetto di Nash equilibrium e' troppo ampio o troppo restrittivo? La mia lettura della letteratura e' che sia troppo ampio, da cui i numerosi giochi con Nash equilibria anche apparentemente paradossali, i vari refinement, ecc.. Invece, leggendo quello che hai scritto, ho avuto l'impressione che tu pensi che sia troppo restrittivo. E' cosi'?
Mailath G., "Do People Play Nash Equilibrium? Lessons From Evolutionary Game Theory", Journal of Economic Literature, 36 (September 1998), 1347-1374
In particolare c'e' una sezione su Nash Equilibrium as an Evolutionary Stable State, che forse si avvicina a cio' a cui stai pensando tu.
Una cosa pero' ancora mi sfugge. Secondo te, il concetto di Nash equilibrium e' troppo ampio o troppo restrittivo? La mia lettura della letteratura e' che sia troppo ampio, da cui i numerosi giochi con Nash equilibria anche apparentemente paradossali, i vari refinement, ecc.. Invece, leggendo quello che hai scritto, ho avuto l'impressione che tu pensi che sia troppo restrittivo. E' cosi'?