Frames and social games

The Journal of Socio-Economics 45 (2013) 227–233

Contents lists available at SciVerse ScienceDirect

The Journal of Socio-Economics journal homepage: www.elsevier.com/locate/soceco

Frames and social games Diego Lanzi ∗ Faculty of Economics, Alma Mater Studiorum - University of Bologna, Italy

a r t i c l e

i n f o

Article history: Received 12 September 2011 Received in revised form 31 July 2012 Accepted 23 August 2012 JEL classification: C70 C71 Keywords: Social games Frames Framing effects Erving Goffman

a b s t r a c t In this paper, we model socially-embedded games. We use non-cooperative game forms with pure strategy Nash equilibria and embed them through framing structures. These frames alter how players perceive the game, or rule out the choice of some elements of players’ option sets. In this way, we explicitly link the notion of social game to concepts taken from Erving Goffman’s theory of interaction. According to Goffman, game theory is flawed because it applies a single-level model to two-leveled situations, ignoring the fact that people form impressions and expectations during or before any strategic game. In this essay, we provide a set-up endowed with these two levels. Firstly, players endogenize a given setting and frame the interaction. This determines what kind of game they will play. Secondly, they select Nash equilibrium strategies. As we shall discuss, it is possible in this way to consider how frames operate and what role they have in determining Nash solutions of non-cooperative games. © 2012 Elsevier Inc. All rights reserved.

1. Introduction During the last twenty years several books have provided alternative syntheses of what economists consider useful of contemporary game theory.1 A common feature of these contributions is that they contain few references to the “social dimension” of strategic interaction. For instance, according to the wellestablished Arrowian tradition, in the impressive 639-page volume “Playing for real” by Ken Binmore, a master in game theory, the adjective “social” almost always is matched with “welfare functions” or “decision rules”. Indeed, this restrictive interpretation of the social nature of strategic interaction makes game theory sociologically-reductionist and endorses an excessively individualistic notion of interaction order. For instance, take the concept of Nash equilibrium (Nash, 1951). As is well known, Nash suggests that the interaction order results from the correspondence of individual best responses, but he ignores the fact that, in many cases, this order is impossible without non-individualistic categories, mechanisms or constructions. Moreover, as the work of Thomas Shelling has clearly anticipated (see Shelling, 1960), even the kind of order an interaction achieves could be dictated by social factors, normative principles or interpersonal conventions. Neoclassic economists, however, prefer to suppose that social influences simply shape

players’ preference ordering over game outcomes. Impressions, relational status, social pressures and social stigma do not play any role in their theories. Methodologically, there are good reasons explaining why social structures, social roles and socially-determined norms of behavior have been widely neglected by economic science. To inject social structures in games would have driven early game theory too far from the realm of intentional action, and strategic interactions in which actions and outcomes are only loosely determined by individual decisions and dispositions fit badly with Von Neumann and Morgenstern’s definition of game. Only very recently, some contributions have tried to enlarge the game theorists’ toolbox in order to reduce its social ingenuity. Refinements have been proposed from different perspectives.2 Some papers deal with non-cooperative games considering an endogenous interaction structure and players who simultaneously choose actions and players. In Hojman and Szeidl (2006) individual behavior is embedded in a social network, deliberately created by the players, and a social game is conceived as a triplet of elements: a finite number of agents, an underlying game structure (i.e. a traditional non-cooperative game), and a function which determines the cost of links with other players. Similarly, Jackson and Watts (2005) define a social game as a triple: a finite number of players,

∗ Tel.: +39 0512098888; fax: +39 0512098040. E-mail address: [email protected] 1 Among others: Fudenberg and Tirole (1991), Binmore (1992), Obsourne and Rubinstein (1994) and, more recently, Binmore (2007).

2 In fact, the first attempt to consider the sociality of games can be found in Aumann (1974). His notion of correlated Nash equilibrium is based on the idea that players might build “signalling devices” to coordinate towards payoff-improving equilibria. Roughly, Aumann conceives sociality as interpersonal co-ordination.

1053-5357/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.socec.2012.08.003

228

D. Lanzi / The Journal of Socio-Economics 45 (2013) 227–233

an underlying game structure and a set of roles which determine role-oriented partitions of the reference population of agents (for instance, males and females). Both contributions determine refined Nash equilibria in which, respectively, players have to build an equilibrium network structure (Nash network) or have to efficiently match one with the other. Bervoets (2007) follows a different research direction. He shows a way of embedding games in social contexts using game forms.3 Given a finite population of agents (namely, a society), a finite set of strategies for each agent and an outcome function, Bervoets approaches the issue: “In which society does an individual enjoy more options?” even if, in this way, he ends up viewing the social embeddedness of games as formal interdependency in the definition of socially-accepted players’ rights. These rights determine what kind of behavior will be socially permitted, and which behavioral options will be granted by the social context. Finally, Herings et al. (2007) generalize the idea of cooperative, non-transferable utility games and define a socially-structured game. The latter is a cooperative interaction with a given set of agents, a finite set of admissible internal organizations between members of coalitions of players, and a power function determining the distribution of power between members of the admissible organizations (more powerful players are able to remove payoff units from less powerful ones). Herings et al. state that a payoff vector is socially stable if and only if a collection of coalitions and internal organizations which balance players’ power exist. Not surprisingly, these authors provide conditions for socially-stable, core allocations of resources.4 In this paper, we explore a new route for modeling socially structured games. Firstly, like Bervoets (2007), we use game forms. Secondly, we explicitly model, as in Herings et al. (2007), social structures (even if we interpret them from a different sociological viewpoint (see below)). Thirdly, following Jackson and Watts (2005) and Hojman and Szeidl (2006), we confine our reasoning to noncooperative, normal form games with perfect information and pure strategy Nash equilibria. Two remarkable differences between the above papers and our model exist. On the one hand, following contemporary sociology, we view social structures as webs of social values, moral dictates and normative principles. These embeddedness structures alter how players perceive the game, and alter the possibility of choosing some elements from players’ option sets.5 Furthermore, these structures can be intentionally activated or unintentionally internalized. On the other hand, we explicitly link social games to concepts taken from Erving Goffman’s theory of interaction.6 As is well known, Goffman explains interaction order in terms of situational normality, inter-subjective consensus and reflexive expectations, and uses the concept of frame to explain the manner in which social interactions are defined and perceived by interactants. During 1970s his ideas about how a game operates were in open contrast with the orthodox view and Goffman was one of the most visible critics of early game theory. His approach was radical: game theory was flawed because it applied a single-level model to

3 The idea of game form was originally introduced by Sen (1970). For details on this notion, and its uses in social choice theory, see Gibbard (1973) and, more recently, Gaertner et al. (1992). 4 Burns and Gomolinska (2000) suggest another interesting way of modelling socially-embedded games. They use the theory of rule complexes. Unfortunately, social games they use are not comparable with traditional definitions of a game. For this reason, their ground-breaking contribution has not received wide attention. 5 We take the idea of embeddedness from the seminal works of Granovetter. See Granovetter (1985, 2005). 6 See Goffman (1959, 1963, 1969, 1974, 1983). For comments and criticisms to Goffman’s ideas see, inter alia, Garfinkel (1967), Jameson (1976), Maynard (1984, 1991), Cahill (1998) and Misztal (2001).

two-leveled situations, thus ignoring the fact that people manipulate impressions and expectations before, and during, strategic games. This is a venial sin, if we deal with fun games; it becomes an important lack when we analyze socially-embedded interactions. Starting from Goffman’s heritage, we shall provide a model endowed with these two levels. Firstly, players endogenize a given social structure and frame the interaction. This determines what kind of game they will play. Secondly, they select Nash equilibrium strategies. As we shall discuss, it is possible to consider, in this way, how framing structures operate and what role they have in determining Nash solutions. What we get at the end of our reasoning will be a Goffman-inspired theory of social games. The essay is organized as follows. In the next section, we introduce our set-up and provide definitions and notation. Then, in Section 3, we show the use of our framework in a famous 2X2 normal form game. Some introductory results are presented in Section 4, while Section 5 discusses them and presents a simple application. The last section is reserved to a concluding comment. 2. Definitions and notation Consider a finite population of players (i = 1, . . ., n) and denote with C what we shall call a game compact. This compact is a set of possible, alternative, descriptions of a game-like situation which is going to interest the above agents. As we shall see, the final form of the game will depend on what framing structure embeds the interaction. More precisely, C specifies players’ option sets (i with i = 1, . . ., n) and, therefore, describes alternative socializations of a game. Option sets contain sets of permitted strategy sets (Si ), one for any feasible description of the strategic interaction.7 Formally: i := {∀Si |Si ∈ C} ∀i

(1)

A strategic interaction is described as a game form (). The latter is given, as usual, by two elements: the Cartesian product n of players’ S → X, i.e. current strategy sets and an outcome function, g : i=1 i a map which associates a game outcome (x ∈ X) to any n-tuple of individual strategies.8 In order to obtain individual payoffs, we consider a value function, V : X → R+ . This determines a social surplus value for any possible game Finally, we shall denote with yi individual outcome. n y = V (x), x ∈ X and yi ≥ 0 ∀ i. Consistently, a game payoffs with i i=1 outcome will be collectively optimal only if V(x) ≥ V(x ) ∀ x = / x. Let us proceed to introduce some social features into this set-up. Firstly, suppose that some values are seen as relevant for characterizing the game compact we are dealing with. For the sake of tractability, assume they are common to all players and denote them vj with j = 1, . . ., J. An option map, i : i → R+ , sets out how different feasible strategy sets perform in terms of value fulfillment. Straightforwardly, by using i , the bliss set can be defined as: Si∗ := {Si |Si ∈ C ∧ Si ∈ C s.t. i (Si ) > i (Si∗ )}

(2)

Moreover, in order to compare different game outcomes in terms of value fulfillment, players use a evaluation map, mi : X → R+ , that is an outcome-oriented version of i . Namely, the couple of elements i : = (i , mi ) is said to be the values system of each interactant in C.9 After all this, we are equipped to provide some important definitions. Definition 1. A meta-game  is a couple of elements n  i=1 Si , g , < i >i=1,...,n .

7

By assumption, singletons are not allowed. Maps and functions we introduce hereafter are assumed to be twicecontinuously differentiable. 9 Hereafter, we suppose that mi and i are additively separable. 8

D. Lanzi / The Journal of Socio-Economics 45 (2013) 227–233

Definition 2.

A strategy profile s∗ ∈

efficient Nash equilibrium (SENE) of 

n

S ∗ is i=1 i if s ∈

229

a socially-

 n  Si ∈C ( i=1 Si )

∗ ))) +  (  ∗ ∗ such that yi (V (g(si , s−i i Si ) + mi (g(si , s−i )) ≥ yi (V (g(s ))) +

Si = / Si∗ . i (Si∗ ) + mi (g(s∗ )) ∀i and ∀ Definition 3.

A strategy profile s◦ ∈

n

S◦ i=1 i

n

Intuitively, a meta-game specifies a game form and a set of possible, alternative, descriptions of the strategic interaction at stake. Consistently, a socially-efficient Nash equilibrium is a profile of strategies such that there is mutual non-deviation with respect to all the alternative strategy profiles of any description of the game (e.g. for any selection of Si ) . Such a solution has the traditional properties of a Nash equilibrium, given that players have no reason to deviate from it, but it demands that current strategy sets coincide exactly with bliss ones. When this does not happen (i.e. Si◦ = / Si∗ ), we have a socially-constrained Nash equilibrium. In this case, value satisfaction is not maximized and Nash equilibria of this sort are constrained because it should be possible, at least in principle, to modify the way the interaction is constructed to increase social value attainment levels.10 The next definition suggests how to order different Nash equilibria associated with alternative descriptions of a game-like situation. Definition 4.

For any Si1 , Si2 ∈ i , N i is a Nash order relation

∗,1 ∗,1 on i such that if yi (V (g(si∗,1 , s−i ))) + i (Si1 ) + mi (g(si∗,1 , s−i )) ≥

∗,2 ∗,2 2 ∗,p yi (V (g(si∗,2 , s−i )) + i (Si2 ) + mi (g(si∗,2 , s−i )) then Si1 N i Si with s denoting Nash equilibrium strategy profiles for p = 1, 2.

In what follows, we assume that this order relation is complete, transitive, reflexive and antisymmetric on i . By using N i individual option sets can be handled as partially-ordered sets (posets) and we can introduce a tool borrowed from posets mathematics: morphism.11 Let (i , N i ) be a finite and non-empty poset. Assume that i has finite length l with l := max d(, Si ) + 1

(3)

where  is a maximal element of i . In this case, we can decompose i into an K-tuple (L1 , . . ., LK ) of levels. Each level is a set defined as follows: Lk := {ϒi ⊂ i | ∀Si ∈ ϒi rk(Si ) = c} with c > 0

(4)

where we assume that a rank function, rk : i → I+ , exists such that rk(Si ) > rk(Si ) if d(, Si ) < d(, Si ).

i be two posets. A map ϕ : i →  i is said to Thus, let i and  be order-preserving if:

N    Si N i Si in i then ϕ(Si )i ϕ(Si ) in i

(5)

or, alternatively, when rk(ϕ(Si )) = rk(Si ) for ∀Si ∈ i

Fig. 1. Hawk–Dove game.

is a socially-

constrained Nash equilibrium (SCNE) of  if s ∈ ( i=1 Si◦ ) ◦ ))) +  (S ◦ ) + m (g(s , s◦ )) ≥ y (V (g(s◦ ))) + such that yi (V (g(si , s−i i i i i i −i ◦ ◦ i (Si ) + mi (g(s )) ∀i and with Si◦ ∈ C, Si◦ = / Si∗ .

A one-to-one, and onto, order-preserving map is generally named isomorphism. By contrast, we call allomorphism a non-order-

i such that: preserving map : i → 

rk( (Si )) = / rk(Si )

for at least one Si ∈ i . Allomorphism will be used for modeling the manner in which framing structures modify the relative desirability of permitted strategy sets. Algebraically, these morphisms operate as functional transformations of (mi )ni=1 and/or (i )ni=1 which determine the emergence of the binding description of the game.12 We shall call a framing morphism. Finally, the definition of social game. A social game is a metagame on which framing morphisms operate and select a binding game form. In other words: social games are here conceived as framed meta-games. As we shall appreciate, this definition fits well with several elements of Erving Goffman’s interaction theory. This allows us to connect social games with his line of thought. 3. A meta-game Can the decision of being, or not, aggressive be the result of framing activities between interaction participants? Is it possible that, by differently framing a game, players can transform a strategic interaction? According to Goffman, the answers to the above questions are affirmative. In this section we illustrate how this transformation can be reconstructed. Let us consider two versions of the Hawk–Dove game named, respectively, Selfish Killers game and Gangsta Paradise game (see Fig. 1). The Selfish Killers game presents the case of self-interested individuals who fail to be coordinated to achieve a collectivelyoptimal outcome. Contrarily, the Gangsta Paradise game considers two altruistic players who, by forgoing individual rewards, are able to reach the socially-optimal solution of the game. In the first case, Hawk (h) strictly dominates Dove (d); in the second, the opposite holds. In other words, selfish individuals always prefer being Hawks, while altruistic ones behave like Doves. Indeed, from a Goffmanian point of view, some ways of transforming the first game in the second (or vice versa) exist. By using our set-up, let us illustrate how to handle the issue. The game we are dealing with is a two-player, normal form, game for which two alternative descriptions exist. In this very simple example, individual option sets are identical and given by: i = {Hi , Di } ∀i with Hi ≡ Di = {d, h}

10 Note that a socially-efficient game outcome is a strategy profile which maximizes the degree of satisfaction of social values. Hence, in our set-up, it is possible to have collectively optimal, but socially-inefficient Nash equilibria. 11 For an introduction to poset mathematics see Davey and Priestley (1990) and Rival (1982).

(8)

Since Hi ≡ Di , it is true that i (Hi ) = i (Di ). Furthermore, the evalutation maps for any player here are of the kind: mi (h, h)|Hi > mi (d, d)|Hi

(6)

(7)

(9)

mi (d, d)|Di > mi (h, h)|Di Expression (8) says that different ways of keying the interaction cannot modify individual strategy sets. This is a special case: alternative descriptions of a strategic interaction can alter individual

12

We shall denote the transformed versions of these maps with m and  .

230

D. Lanzi / The Journal of Socio-Economics 45 (2013) 227–233

strategy sets. Here, however, this does not happen and, therefore: (i) socially-constrained Nash equilibria are ruled out from the reasoning and (ii) framing morphisms must be simple sub-frames. Furthermore, the two Nash equilibria, (s1 , s2 ) = (h, h) for selfish killers and (t1 , t2 ) = (d, d) for altrustic gagsters, can be described, in our notation, as follows: g(s1 , s2 ) = (h, h); V (g(s1 , s2 )) = 0; y1H = y2H = 0;

g(t1 , t2 ) = (d, d) V (g(t1 , t2 )) = 2

(10)

y1D = y2D = 1

With this definition of the mate-game in the hands, consider two conflicting social values, say others-oriented (e.g. social loyalty) and self-oriented one (e.g. individual utility). Consistently, we can distinguish two cases: (i) yiH + mi (g(s1 , s2 )) ≤ yiD + mi (g(1 , 2 ))

H D with Di N i Hi ∀i and (ii) yi + mi (g(s1 , s2 )) ≥ yi + mi (g(1 , 2 )) with

Hi N i Di ∀i. In the first case, social motivations sustain altruistic behavior. Players are perfectly altruistic and the Nash equilibrium of the game is socially-efficient and collectively-optimal. However, in the second case, behavioral norms sustain non-cooperative and self-interested choices and the socially-efficient Nash equilibrium is not collectively-optimal. So, alternative norms can make sociallyefficient different game outcomes and manipulating these value elements may alter the way in which the meta-game is perceived. For instance, members of an organized crime clan (or populations of areas with a high density of gangs members) internalize particular norms of behavior (for instance, silence, distorted ideas of self-respect, and many group-related stereotypes) which can transform an otherwise-inefficient behavior into a socially-optimal line of action.13 Therefore, as gangsters well know, a prisoner dilemma can be, in certain social contexts, a real delight. But, what happens when our social values coexist? Now, how players consider game outcomes will depend on the importance assigned to each social value. In the case of equal weighting, a cooperative solution of the Hawk–Dove game should emerges as a sociallyefficient Nash equilibrium. Non-cooperative behavior is the case only with different weighting between values. Framing structures may intervene precisely in this case: if our selfish killers have seen and suffered too many murders and violent acts they could develop strong aversion towards clan rules, re-frame the game, reject clan values and they may not select a collectively-optimal outcome.14 Hence at least a frame, which makes the two versions of the game equivalent must exist. Take, for instance,



such that yiH + mi (g(s1 , s2 )) = yiD + mi (g(t1 , t2 )). If the last condition holds, we have two socially-equivalent descriptions of the interaction. 4. Social games and frames: some introductory results The above discussion illustrates how two alternative takes of the Hawk–Dove game can be compared. They are, in fact, the same meta-game, framed in different meaning contexts. For the sake of our argument, the central point of the example is twofold. On the one hand, it is possible to transform games. On the other hand, interactants can drive the Hawk–Dove meta-game towards one of its alternative descriptions. Consistently, in this section, we provide some results which explain, in more general terms, the idea of social equivalence between games and show how framing activities may shape social games.

13 The term social must refer here to clan members. For an example about what happens in the South of Italy to mafia clan members. See Gambetta (1988). 14 This is the so called Buscetta case. On this point see Gambetta (1988).

Let us consider two game forms with n players, say  and  , and our definition of social game. Formally, the social equivalence of game forms can be written as follows: Definition 5. Given a meta-game , two game forms (Pn , g) and  n , g), are socially-equivalent only if rk(P ) = rk(S ) ∀ i and p∗ ∈  i i (S n n ∗ ∈ 15 P , s S are socially-efficient Nash equilibria. i i i=1 i=1 According to Definition 5, socially-equivalent game forms have two important features: (i) they ensure the same level of satisfaction of binding social values; (ii) they reach socially-efficient game outcomes. As stressed above, we shall investigate the role of framing structures, or other similar individual dispositions, in transforming a game into a socially-equivalent one. J Let (vj )j=1 be a bundle of social values and N i a Nash order relation. Moreover, let be a framing morphism. By using our framework, we can show the following: Proposition 1.

Given  and (vj )j=1 , ∀(,  ) ∃ s.t.    . J

Proof. Take two forms, say  and  , and suppose that p∗ ∈  game n n ∗

P and s ∈ i=1 Si are their socially-efficient Nash equilibria. i=1 i Consider, without loss of generality, the first game. If s∗ is a sociallyefficient Nash equilibrium, it must be that g(s∗ ) = x∗ ∈ XSN = / g(p∗ ). Moreover, the following condition must hold: yi (x∗ ) + i (Si ) + mi (x∗ ) = max(yi (x) + i (Si ) + mi (x))

∀x = / x∗ ∈ XSN and ∀i If

g(p∗ )

(11)

= z ∈ XSi , it is true that:

yi (z) + i (Pi ) + mi (z) < yi (x∗ ) + i (Si ) + mi (x∗ )∀i / XSi . and that rk(Pi ) < rk(Si ) ∀ i. Therefore, it must be that g(p∗ ) = z ∈ Use now an allomorphism such that ∀i rk( (Pi )) = rk(Si ) rk( (Si )) = rk(Pi )



yi (z) + i (Pi ) + mi (z)

(12) =

yi (x∗ ) + i (Si ) + mi (x∗ )

J

Given (vj )j=1 , and according to Definition 4, if operates then    . Proposition 1 generalizes the intuition laying below the example of Section 3. The morphism alters values satisfaction systems and modifies the relative goodness of strategy sets. Alternative ways of socializing an interaction lead to different games and different socially-efficient outcomes. Moreover, it is always possible, given a meta-game, to re-frame the interaction for getting a sociallyequivalent game form with different, but still socially-efficient, Nash equilibria. Until now, however, we have assumed two game forms with the same family of binding social values. Using the expression introduced by Burns and Gomolinska (2000), we discussed of closed games in which reference values cannot be modified. Contrarily, in open social games social values can be changed. Proposition 2 will deal with this, more general, class of games. In particular, when social values change socially-efficient Nash equilibria can become socially-constrained ones through a framing structure which forges a different social ontology. Consistently, the next Proposition illustrates which relation occurs between sociallyconstrained and socially-efficient Nash equilibria in open games. Proposition 2. Given , ∀(,  ) with (vj )j=1 ∃[ , (vj )j=1 ] s.t.    and any SCNE of  becomes a SENE of  (and vice versa). J

J

15 For the sake of exposition, the writing    will mean that the two game forms are socially-equivalent.

D. Lanzi / The Journal of Socio-Economics 45 (2013) 227–233 J

Consider some initial social values (vj )j=1 and the

Proof.

n

meta-game  . Let s◦ ∈ i=1 Si◦ be the socially-constrained Nash  . Then, there exists a strategy profile, say s∗ ∈ equilibrium of  Si∗ ∈



Si ∈C

Si ) with Si∗ = / Si◦ such that:

(

i

yi (V (g(s ))) +

i (Si∗ ) + mi (g(s∗ ))

◦

> yi (V (g(s )))

with

>

i (Si◦ ) ∀i.

(13)

Suppose now that frames the game and

J that a new bundle of relevant social values is identified (e.g. (vj )j=1 ).

By using the following expressions:

= rk(Si◦ )



yi (V (g(s◦ ))) + i (Si◦ ) + mi (g(s◦ )) +mi (g(s∗ ))

(14)

= yi (V (g(s∗ ))) + i (Si∗ )



Si∗ ∈ /





Si

Si ∈C

i

a

i

SENE of  is a SCNE of the socially-equivalent game form  .  Through Proposition 2 we can appreciate the difference between a socially-constrained Nash equilibrium and a socially-efficient one. The former is a Nash equilibrium of a socially-inefficient closed game. Even if interaction participants perceive social values, or their assessment systems, as inadequate they cannot reject them and look for new game modalities. Differently, if a game is open players can detect new values, revise the socialization of the interaction and re-frame it. This may transform sociallyconstrained equilibria into socially-efficient ones. Similarly, if we have a socially-efficient Nash equilibrium of an open game, it is possible to re-frame the interaction and exclude (Si∗ )i∈N from players’ option sets: interactants’ response will be to select a sociallyconstrained Nash equilibrium. Finally, framing morphism can be used to convert collectivelyoptimal outcomes, which do not correspond to equilibrium profiles, in socially-efficient Nash equilibria. The next Proposition illustrates how this happens for closed games. J

Proposition 3. Given  and (vj )j=1 , ∃ s.t. collectively-optimal outcomes of  become socially-efficient Nash equilibria.



Si◦ is a socially efficient Nash equi-

i

librium and that x◦ = g(s◦ ), the outcome associated with such an equilibrium is not collectively optimal. According to Definition 2, the following expression must hold: yi (V (g(s◦ ))) + i (Si◦ )+mi (g(s◦ )) ≥ yi (V (g(s )))+i (Si ) + mi (g(s )) ∈





i

i

Si = /

Si◦i, ∀

(15)

By considering all agents, if condition (14) is verified then it must be that for any s : yi (V (g(s◦ ))) +

i



i (Si◦ ) +

i

+

i



mi (g(s◦ )) ≥

i

i (Si ) +

i



(17)

yi (V (x◦ ))

i

Therefore, the next condition must hold:



mi (g(s◦ )) −



i

mi (g( s)) >



i

yi (V ( x)) >

i



yi (V (x◦ ))

(18)

i



mi (g(s◦ )) −



s∈ If (19) holds, 

mi (g( s)) < 0

(19)

i



Si◦ becomes the new SENE of . 

i

that also the opposite holds. Finally, if

∀s

yi (V ( x)) >

i

we can characterize the game form  such that    and in which s◦ is a SENE. Following the same line of reasoning, it easy  to show



V ( x) > V (x◦ ) and

Now, it is sufficient to take such that ∀i

rk( (Si∗ ))

Suppose that s◦ ∈

Si◦ such that:

i



rk( (Si◦ )) = rk(Si∗ )

Proof.

least an outcome, say  x = g( s) with  s∈

i

+ i (Si◦ ) + mi (g(s◦ ))∀i i (Si∗ )

there must exist at But then, since x◦ is not collectively optimal, 



i ∗

231



yi (V (g(s )))

i

mi (g(s ))

(16)

5. Discussion and application The core of above propositions lies in the fact that human beings have the capacity to reject, redefine, transform and negotiate reality. Sometimes they face social interactions which cannot be manipulated, or re-framed, but, in the vast majority of games, identities, roles and reference values can be altered deliberately. By modifying framing structures, interaction participants change their roles and the way in which the interaction or the plot of the game is perceived. In particular, given a set of social values (i.e. a closed game), agents can negotiate which weight to assign to each value and the relative “goodness” of different strategy sets. When such a transformation process succeeds, interactants forge a new game form which is different, but socially-equivalent to the primary one (Proposition 1). Following contemporary sociological theory, there are three main ways to re-frame an interaction (see, among others, Schmitt, 1985): (i) positive framing, i.e. interactants are fully aware that reality cannot be as they would wish, but a mutual-pretence-awareness-context is built up and defended; (ii) negative framing, i.e. interaction participants construct an overtly-ambiguous-awareness-context which operates as a bidding non-reality; and, (iii) fabricating, i.e. each player fabricates a particular re-construction of reality and any interactant faces a closed-awareness-context in which he/she cannot reveal the deception tactics of the others. In the meta-game of Section 3, players might use, for instance, a positive frame to transform a dilemma into a delight. In closed games, interaction participants can also find wellshaped framing morphisms to solve existing conflicts between social and collective rationality (Proposition 3). This means that players can describe differently the interaction, and fabricate different ways of surviving in the game. Open games are a little more complex. In these strategic interactions, not only frames are at work, but also shifts in the reference social values take place. By ruling out the relevance or legitimacy of some values, interactants can change the composition of individual option sets and hence their possible patterns of behavior. In this case, players are giving life to a new social ontology and, by manipulating several causal mechanisms (like beliefs, emotions, memory, desires, sense of self, personal, social identity, etc.), they are building a completely new game. When this happens, socially-constrained Nash equilibria can be transformed into socially-efficient ones, and vice versa (Proposition 2). In the first case, the social ontology is creating enlarged option sets; in the second one, options shrink and some modes of behavior are simply not possible.

232

D. Lanzi / The Journal of Socio-Economics 45 (2013) 227–233

Moreover, doing the same for 2 permits us to establish that: yi (a, a) − yi (b, b) = 5 (24)

mi (a, a) > mi (b, b)

Since (a, a) is socially-efficient in 2 , conditions for the social efficiency of (a, a) in 1 must be determined. To do this, introduce a morphism and apply it to the poset (i , N i ). If is such that: rk( (Si1 )) = rk(Si2 ) rk( (Si2 )) = rk(Si1 )



i (Si1 ) + mi (a, a)

Fig. 2.

To give substance to the above comments, let us deal with a simple social game. Suppose we have two players (˛, ˇ) and consider, as we have done from the beginning of this paper, only pure strategies. Individual option sets are supposed to be as follows: i = {Si1 = (a, b, c), Si2 = (a, b), Si3 = (b, c)} for i = ˛, ˇ

(20)

Through (20), we can distinguish three different game forms: 1 , 2 and 3 that, for the sake of tractability, are represented here by the above matrices (see Fig. 2): 1 has two Nash equilibria: (a, a) and (c, c). The first is collectively optimal; the second is collectively disruptive. Obviously, if interactants cannot negotiate the manner in which the interaction is socialized, and have to stick to 1 , they will face a pure-coordination game. The second game form, which emerges when players consider as not acceptable strategy c, is a purecoordination game as well. This game has two Nash equilibria: (a, a), which is one of the Nash equilibria of 1 , and (b, b), that was not an equilibrium in 1 . Finally, 3 has a unique equilibrium in dominant strategies, (c, c), and this outcome is both collectively and individually sub-optimal. Now consider two reference social values, say v and v◦ , and take i and mi such that ∀i i (Si2 )

>

i (Si1 )

>

i (Si3 )

mi (a, a) > mi (b, b) > mi (c, c)

(21)

According to our set-up, (a, a) ∈ 1 is collectively-optimal and Si∗ ≡ , we get the following ordering: Si2 ∀i. Hence, by using N i S 1 N S 3 ∀i Si2 N i i i i

(22)

Moreover, Definitions 2 and 3 allow us to detect the properties of the above Nash equilibria. The strategy profile (a, a) is the sociallyefficient equilibrium of 2 , but it becomes a socially-constrained outcome of the first game in Fig. 2. The game outcome (c, c) is the socially-constrained Nash equilibrium of 1 and 3 , while the couple (b, b) is the socially-constrained equilibrium of 2 . We now have all the elements we need to illustrate our propositions. Let us start with Proposition 1, that is, it is possible to find a framing morphism that makes socially-equivalent two game forms. Consider, as an example, (1 , 2 ) and (2 , 3 ).16 With reference to the first couple of elements, take 1 and compare its Nash equilibria; it can be easily checked that: yi (a, a) − yi (c, c) = 7 mi (a, a) > mi (c, c)

16

For the sake of brevity, the remaining cases are omitted.

(23)

(25) =

i (Si2 ) + mi (a, a)

the couple (a, a) becomes a socially-efficient equilibrium of 1 and 1  2 . Slightly different is the case of (2 , 3 ). Following what we have done above, we can write: yi (a, a) + i (Si2 ) + mi (a, a) > yi (c, c) + i (Si3 ) + mi (c, c)

(26)

In this case, in order to ensure that (c, c) is socially-efficient in 3 and 2  3 , we need a framing morphism such that: rk( (Si3 )) = rk(Si2 ) rk( (Si2 )) = rk(Si3 )

(27)



i (Si3 ) + mi (c, c) ≥ 7 + i (Si ) + mi (a, a) Proposition 2 has dealt with the relation which exists between socially-efficient and socially-constrained Nash equilibria. Roughly, it suggests that any open game can be re-framed to make sociallyefficient equilibria that are socially-constrained (and vice versa). To gain insights into the last point, consider the Nash equilibrium (a, a), 1 and 2 . Given some social values, the socially-efficient equilibrium profile in 2 is socially-constrained in 1 . Therefore, according to Proposition 2, we have to modify our reference values and re-frame 2 . To do this, take (v , v0 ) and with

(v , v0 )|i ( (Si1 )) + mi (a, a) ≥ i (Si2 ) + mi (a, a)

(28) 2

becomes When (28) holds, the socially-efficient equilibrium of socially-constrained and the socially-constrained equilibrium of 1 becomes socially-efficient. Finally, Proposition 3 stresses that whenever a collectively-optimal outcome is not a socially-efficient Nash equilibrium, we can find a framing morphism which ensures the correspondence between the two. In our example, (a, a) ∈ 1 is collectively-optimal, but not socially-efficient. By following the reasoning outlined above, we can identify a framing morphism such that (a, a) is both socially-efficient and collectively optimal. To get the result, it is sufficient to take such that: rk( (Si1 )) = rk(Si2 ) rk( (Si2 )) = rk(Si1 )

mi (a, a)

>

(29)

mi (b, b)

6. A concluding comment In this paper, we have applied Erving Goffman’s view that social “situatedness” has important consequences on the way in which strategic interaction operates. In particular, we have introduced the notion of social game, which makes possible to formalize some elements (like framing structures) which, according to Goffman (1964, 1984) and several discussants of his ideas17 are relevant for

17

See for instance Maynard (1991).

D. Lanzi / The Journal of Socio-Economics 45 (2013) 227–233

determining the interaction order. Consistently, we have focused on the ways in which players can breach, and thereby exploit, different socializations of a game-like situation. Erving Goffman has also emphasized the relevance of processes and practices aimed at establishing “rules of the game” through the creation of social meaning. This creation requires the selection of social values and the development of value systems, as well as intentional dispositions in favour of a relevant framing structure. Following these insights, the notion of frame allows us to develop an embedded interaction theory.18 Furthermore, Goffman thought that “too much is left out” of game theory (see Goffman, 1983). Hence, if we agree with this statement, it is essential to enrich the theory with some of Goffman’s suggestions and see what happens. As we have shown, contemporary game theory and Goffman’s sociology are not separate entities and can be fruitfully intertwined. Such a reciprocal fertilization gives us a richer understanding of the social significance of games. In this perspective, frames and social values are the first things to be brought into game theory. Indeed, in the last decades, sociology has made widespread use of games. Sociological versions of game theory have dealt with social dilemmas or with structural, or motivational, solutions to game-like situations.19 However, this literature has not been able to develop an independent notion of socially-embedded games. Any game defined by game theory is social for sociologists, but, unfortunately, this position eludes the need for a formal analysis of the social bases of games. On the other hand, economists have proposed notions of social game, and, in this respect, have been more creative and open-minded than their sociology colleagues. However, few economists understand social theory and, hence, economists’ notions of social game are sociologically weak. To avoid this paradoxical situation, the contributions of Erving Goffman are brilliant starting points. References Aumann, R., 1974. Subjectivity and correlation in randomized strategies. Journal of Mathematical Economics 1 (1), 67–96. Bervoets, S., 2007. Freedom of choice in a social context: comparing game forms. Social Choice and Welfare 29 (2), 295–315. Binmore, K., 1992. Fun and Games. Heath and Co., Lexington, MA. Binmore, K., 2007. Playing for Real. Oxford University Press, Oxford. Burns, T.R., Gomolinska, A., 2000. The theory of socially embedded games: the mathematics of social relationships, rule complexes, and action modalities. Quality and Quantity 34, 379–406.

18 Lanzi (2010, 2011) suggest an embedded choice theory. Hence, this paper can be seen as a game-theoretic complement to those contributions. 19 For a survey on the main uses of game theory in sociology see Swedberg (2001).

233

Cahill, S., 1998. Toward a sociology of the person. Sociological Theory 16 (2), 131–148. Davey, B.A., Priestley, H.A., 1990. Introduction to Lattices and Order. Cambridge University Press, Cambridge. Fudenberg, D., Tirole, J., 1991. Game Theory. MIT Press, Cam. Mass. Gaertner, W., Pattanaik, P.K., Suzumura, K., 1992. Individual rights revisited. Economica 59 (2), 161–177. Gambetta, D. (Ed.), 1988. Trust: Making and Breaking of Cooperative Relations. Blackwell, Oxford. Garfinkel, H., 1967. Studies in Ethnomethodology. Prentice-Hall, Englewood Cliffs, NJ. Gibbard, A., 1973. Manipulation of voting schemes: a general result. Econometrica 41 (4), 587–601. Goffman, E., 1959. The Presentation of the Self in Everyday Life. Doubleday Anchor, Garden City, NY. Goffman, E., 1963. Stigma. Prentice-Hall, Englewood Cliffs NJ. Goffman, E., 1969. Strategic Interaction. University of Pennsylvania Press, Philadelphia. Goffman, E., 1974. Frame Analysis. Harper, NY. Goffman, E., 1983. The interaction order. American Sociological Review 48 (1), 1–17. Granovetter, M., 1985. Economic action and social structure: the problem of embeddedness. American Journal of Sociology 91 (3), 481–510. Granovetter, M., 2005. The impact of social structure on economic outcomes. Journal of Economic Perspectives 19 (1), 33–50. Herings, P.J., Van der Laan, G., Talman, D., 2007. Socially structured games. Theory and Decision 62 (1), 1–29. Hojman, D.A., Szeidl, A., 2006. Endogeneous networks, social games and evolution. Games and Economic Behavior 55 (2), 112–130. Jackson, M., Watts, A., 2005. Social games: matching and the play of finitely repeated games, FEE working paper n. 38. Jameson, F., 1976. On Goffman’s frame analysis. Theory and Society 3 (1), 119–133. Lanzi, D., 2010. Embedded choices. Theory and Decision 68 (3), 263–280. Lanzi, D., 2011. Frames as choice superstructures. Journal of Socio-Economics 40 (3), 115–138. Misztal, B.A., 2001. Normality and trust in Goffman’s theory of interaction order. Sociological Theory 19 (3), 312–324. Maynard, D.W., 1984. Inside Plea Bargaining. Plenum Press, NY. Maynard, D.W., 1991. Goffman, Garfinkel and games. Sociological Theory 9 (2), 277–281. Nash, J., 1951. Non-cooperative games. Annals of Mathematics 54 (3), 286–295. Obsourne, M.J., Rubinstein, A., 1994. A Course in Game Theory. MIT Press, Cambridge, MA. Rival, I. (Ed.), 1982. posets, NATO ASI Series 83. Reidel, Dordrecht. Shelling, T., 1960. The Strategy of Conflict. Harvard University Press, Cambridge, Mass. Schmitt, R.L., 1985. Negative and positive keying in natural contexts: preserving the transformation concept from death through conflation. Sociological Inquiry 55 (4), 383–401. Sen, A.K., 1970. Collective Choice and Social Welfare. Holden Day, San Francisco. Swedberg, R., 2001. Sociology and game theory: contemporary and historical perspectives. Theory and Society 30, 301–355.