Games in Static Modal Logic

Giacomo Bonanno provides an early example [13] of a modal logic for strategic games. He bases the logic on a Kripke frame _!Ω, R1, ..., Rn, R∗# where Ω is a

set of states,R is a binary relation on Ω,_{1, ..., n_}is the set of players and the (n+ 1)th relation R∗ is the transitive closure of R1∪...∪Rn. However, the

interpretation of the relationRdepends on the interpretation of game theory he takes. One interpretation views “game theory as a description of how rational individuals behave” and the other views it as “a prescription ... to players on how to act.”

Game theory as a description of rational behaviour seeks a way to account for knowledge (of the game and other players) and rationality. Because the stan- dard view of games (in this case, finite non-cooperative strategic form games) “provides only a partial description of the interactive situation” [13], where be- liefs and rationality are not addressed, Bonanno devises a system where they are. In order to “illustrate the types of results obtained” [13] from his formalism, he uses it to describe how a strategy profile survives IEDS in it.

Bonanno adds a probabilistic element to the above frame to account for players’ belief. We get the game model: _!Ω, R1, ..., Rn, R∗, P1, ..., Pn# where Ω

and_{1, ..., n_}are as above. In this interpretation, we specifyRto be associated

with the modal formula !iA which denotes “playeri believes thatA” and is true at a state α _∈ Ω iff A is true at every state β such that αRiβ. The intended interpretation ofR is “for player i stateβ is epistemically accessible from state α” [13]. R∗ is associated with !∗A which denotes “it is common

belief thatA.” Pi is the probability distribution on Ω and is used to describe belief;pi,α which denotes playeri’s belief at stateαis defined by conditioning

Pi onRi(α) ={ω∈Ω :αRiω}:

Definition 4.1.1. Playeri’sbeliefat stateαfor the worldsRi-accessible from

α: pi,α(ω) = # _Pi(ω) ! w!∈_Ri(α)Pi(ω!) ifω∈Ri(α) 0 ifω_'∈Ri(α)

In other words, if ω is in the set of states R-accessible from α, then the probability (belief) ofωforiis a fraction of the total amount of states accessible fromα.

The model also includes a valuation represented by the pair (V, σ) whereV

“associates with every atomic proposition the set of states at which the propo- sition is true” and σ is a function σ = (σ1, ..., σn) : Ω → S “that associates

with every state thepure strategy profile played at that state” [13]. There are two integral atomic propositions that refer to facts about strategy profiles: ri expresses thati is rational, and s∞ is a strategy profile in the set of strategy profilesS∞that survive the process of IEDS. Suppose the following restrictions hold for (V, σ):

1. (a) α_∈V(ri) iffαRiβ⇒σi(β) =σi(α)

(Playerihas no uncertainty of the strategy he himself plays.) (b) σi(α) maximizesi’s expected utility given his beliefs.

2. α_∈V(s∞_{) iffσ(α)∈S}∞

(The strategy played atαsurvives IEDS.)

It is then possible to introduce the formula !∗r →s∞, which claims that “if

there is common belief that all the players are rational, then the strategy profile actually played is one that survives the [IEDS]” [13].

Now we turn to Bonanno’s second interpretation of game theory: as a nor- mative model for which solutions are recommendations to players on how to act. In this interpretation, the meaning of the relationR changes. Here,αRiβ denotes “from state αplayer i can unilaterally bring about stateβ. Thus Ri does not capture the reasoning or epistemic state of player i but rather the notion of what playeriisable to do” [13]. R∗ is interpreted as “at stateαit is

recommended that stateβ be reached.”

In this interpretation, there are also new atomic propositions, wherep, q_∈ Q); (ui = pi) expresses “playeri’s utility is pi,” (q ≤ p) expresses “q is less than or equal to p,” and the proposition Nash expresses “the pure strategy profile played is a Nash equilibrium.” Last, the pair (V, σ) satisfy a new set of restrictions:

1. αRiβ iffσ−i(β) =σ−i(α)

2. Ifais an atomic proposition of the form (q≤p), thenV(a) = Ω ifq≤p

andV(a) =∅otherwise. 3. α_∈V(ui =pi) iffui(σ(α)) =pi

4. α_∈V(Nash) iffσ(α) is a Nash equilibrium of the game.

The modal formulas!iAand !∗A get new interpretations as well. Those are

“no matter what unilateral action playeri takes, A is true” and “it is recom- mended thatA,” respectively. The recommendation to play a Nash equilibrium

is defined as: !∗(

$

(ui=pi)→

$

!i((ui=qi)→(qi ≤pi))→!∗(N ash)

It is generally agreed upon that the strategy profiles in a game are repre- sented as states in a Kripke frame. Bonanno claims that “in order to obtain a model of a particular gameG, [...] we also need to add a function that associates with every state thepure strategy profile played at that state” [13].