Keywords: Kinked Demand, Symmetric Games, Norms of Behaviour,. Coalitions. ... Neither ineffi cient nor asymmetric PE al
Kinked Social Norms and Cooperation Sergio Currarini (U. Venezia Cà Foscari) & Marco A. Marini (U. Urbino "Carlo Bo) September 2007 Abstract We investigate which speci…c social norrns of behauiour - here narrowly interpreted as a commonly shared expectations over people behaviour can sustain a cooperative outcome. We show that in a symmetric setting a speci…c social norm, one shaping the expected. response of all players in the event of individual or coalitional deviations from any collective choice, plays a special role in making.a cooperative outcome stable. Such a norm turns out to be strikingly similar to that assumed in the classical kinkeddemand model by Robinson (1933) and Sweezy (1939) (see also Hall and Hitch (1939)) for …rms operating in imperfectly competitive markets. Keywords: Kinked Demand, Symmetric Games, Norms of Behaviour, Coalitions.
1
Introduction
Often in their social interaction individuals adopt simple behavioural procedures. Social scientists talk to various degree, and with di¤erent meanings, of heuristics, conventions and norms of behaviour. Their emergence can be spontaneous, arising from the evolution of shared expectations into prescriptions and then into norms of behaviour (see, for instance, Lewis 1969, Bicchieri, 1990 and Castelfranchi et a1., 2002). This is a pervasive phenomenon in modern economies and economic actors, as companies’ managers, market traders and CEOs as well as workers and people in general appear similarly inclined to adopt simple procedures rather than complex and elaborate strategies, specially when they have to promptly react to unexpected events. Once established within an organization, i.e.a …rm, a set of norms may constitute its corporate culture.1 Since usually norms are strictly linked to social expectations and then to conjectures individuals form over the behaviour of other people, it is worth to investigate wich form these forces have to take to sustain a cooperative outcome 1 ”The corporate culture acts as a perception …lter, a¤ects the interpretation of information, sets moral and ethical standards, provide rules, norm and heuristics for action, and in‡uences how power and authority are wielded in reaching decision regarding what action to pursue”. (Brown, 1995, p. 197)
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of the economy (i.e. an e¢ cent social outcome) against individual and coalitional deviations. Furthermore, it may be interesting to know whether these norms supporting cooperation are in general rational and if they correspond to some form of reciprocity. A very recent stream of literature has provided strong experimental evidence showing that even in one-shot games with anonymous players people may reward fair behaviour and punish unfair behaviour, even when such a reaction is detrimental to their material payo¤ (see, for a survey, Fehr, Fischbacher and Gachter 2002). Usually this literature classi…es agents’ propensity to support cooperation as i) strong reciprocity, ii) rational altruism or iii) altraism, when, in turn: i) players cooperate conditionally to the cooperation of opponents even when costly; ii) they cooperate only when strictly convenient; iii) they always cooperate, whatever is their expectations over other players’behaviour. In this paper we focus on a simple symmetric and monotone strategic setting and we …nd that a speci…c norm of behaviour - one shaping the expected response of all players in the event of an individual or a coalitional deviation from a collective choice - plays a special role in making the cooperative outcome stable. Such a norm turns to be strikingly similar to that assumed in the classical kinked-demand model by Robinson (1933) and Sweezy (1939) (see also Hall and Hitch (1939)) for …rms operating in imperfectly competitive markets. More speci…cally, we show that for all symmetric games in which such a ’kinked’social norm (KSN) is expected by all players, a Pareto e¢ cient outcome can never be improved upon by any single player or coalition of players. The paper is organized as follows. In the next subsection we present a simple example introducing the main idea of the paper. Section 2 illustrates the setting of the paper. Section 3 presents the main paper results. Section 4 applies these results to the classical kinked-demand model
1.1
An Illustrative Example
To provide an intuition of the logic underlying our results, let us introduce a symmetric two-player game in strategic form in which both players can choose between three strategies, A, B and C, with payo¤s illustrated by the following 3x3 bimatrix:
A B C
A a; a d; f e; h
B f; d b; b l; m
C h; e m; l c; c
Now, let assume that at least a symmetric Pareto e¢ cient outcome (SPE), i.e, an outcome maximizing the sum of all players’payo¤s, exists for this game. As a start, suppose that when every player i = 1; 2 changes from a given strategy 0 xi to a new strategy xi she.expects the the opponent will fully conform to her new choice, i.e., xj = x0i , with j 6= i. It is easy to see that when such a 2
symmetric (ot fully conformist) social norrn of behaviour (SSN) is accepted by all players, this implies that a SPE outcome is the only stable outcome of the game. Neither ine¢ cient nor asymmetric PE allocations can, in fact, by de…nition, give both players a payo¤ greater or equal to the one provided by the SPE outcome. For the same reason, no asymmetric outcomes can be stable against a deviation leading (via the SSN) to the SPE outcome itself. Altough it is not suprising that a full coordination attitude always leads players to a SPE outcome, an important question arises. Is the symmetric social norm the only expected behaviour sustaining cooperation in a symmetric setting? In order to answer this question suppose that all players share a norm of behaviour - that we denote kinked social norrn (KSN) - dictating that when a player i deviates from strategy xi to a new strategy x0i she expect her opponent to respond with a weakly lower strategy xj x0i under positive spillovers (PS) and with a weakly higher strategy xj x0i under negative spillovers (NS).2 To see that such a KSN can support a SPE outcome, assume …rst that players’strategies can be ordered and A > B > C. Let also our game to possess positive spilovers (PS). Therefore, the following inequalities hold: i) a > f > h; ii) d > b > rn and iii) e > :l > c. Assume …rsr that (c; c) ís a SPE outcome. Immagine now that one of the players, say player 1, decides to deviate from this outcome playing A instead of C and, in so doing, she expects the opponent will respònd with a weakly lower strategy. than C, in this case either A, B or C. As a result, player 1 may expect to get either a, h or f , respectively. However, if either a,h or f can improve upon c for player 1, by the PS property a > f > h, and. by the symmetry of the game we obtain that (a; a) can improve on (c; c) for both players, contradicting the fact that (c; c) is a SPE outcome. The same would occur if either (b; b) or (a; a) were a SPE and a deviating player expects her opponent to respond with a weakly lower strategy. Note that the entire reasoning can be repeated under negative spilovers (NS), in which case the KSN would dictate that players are expected to respond to any opponent’s deviation with a strategy weakly lower than deviators.3 Therefore, the above example shows that the stability of a cooperative outcome can be supported by both SSN and KSN. The same cannot be said of a norm implying that players respond with strictly higher strategies than deviators under PS and strictly lower under NS. Suppose, for instance, that (c; c) is a SPE and PS holds. If player l switch from C to B and expects xj > x0i ; the opponent will play A > B and player 1 can easily improve upon c, since there are no constraints in the game imposing that d has to be lower than c. As a …nal remark, we can preliminarly observe that when the game actions are strategic complements (in the sense of Bulow et al., 19S5) and then players’ best replies are increasing, a SPE of the game is stable whenever every player after a deviation expects a rational reponse from her opponent. 2 Note
that the monotone spillover property (ui (xi ; xj ) either strictly increasing or strictly decreasing in xj for every i = 1; 2 and j 6= i) is a very common property of games applied to economic and social analysis. 3 In our example this can be simply done by assuming A < B < C and repeating all steps shown above.
3
2
The Setting
Since the aim of the paper is to …nd social norrns of behaviour (in the narrow interpretation given above) ensuring the stability of a cooperative outcome independently by the peculiar features of the players (as preferences, relative in‡uence or bargaining power) we adopt here a very simple symmetric setting in which players are endowed with the same strategy space and perceive symmetrically all strategy pro…les of the game. To simplify further, we constrain our setting to possess some forms of monotonicity of players’ payo¤s with respect to their opponents’choices. Even so, the resulting setting is still rather general and many well known economic applications (as Cournot and Bertrand oligopoly, public goods games and many others) easily …t into its structure. We denote a a monotone symmetric n-player game in strategic form as a triple GS = N; (Xi ; ui )i2N , in which N = f1; :::; i; :::; ng is the set of players, Xi = X R is every player’s strategy space, assumed compact and convex, ui : X1 ::: Xn ! R+ is the payo¤ function of every player i, assumed continous and strictly quasiconcave, and in which the following two properties hold: P1. (Symmetry) For every i 2 N and for any arrangement of the strategy indexes u1 (x1 ; x2 ; ::; xn ) = u2 (x2 ; x1 ; ::; xn ) = ::: = un (xn ; x2 ; ::; x1 ): P2. (Monotone Spillovers) For every i; j 2 N with j 6= i and every xj > x2j we have either ui (x
1 j ; xj )
(1) x1j
>
> ui (x
j ; xj )
> ui (x
2 j ; xj )
(2)
< ui (x
j ; xj )
< ui (x
2 j ; xj )
(3)
positive spillovers (PS) or ui (x
1 j ; xj )
negative spillovers (NS), where x j = (x1 ; :::; xj 1 ; xj+1 ; ::; xn ): Moreover, since we extend the analysis to strategy pro…les taken by coalitions of players, we denote by S N any coalition of players, and by N nS itsQ complement with respect to N . For each coalition S, we denote by xS 2 XS i2S Xi a pro…le of strategies for players in S N , and use the notation x = xN . Finally, let de…ne a Pareto Optimum (PO) for GS as a strategy pro…le x 2 X n such that there exists no alternative pro…le which is preferred by all players and is strictly preferred by at least one player. The Pareto Optimum xe is e¢ cient (PE) if it maximizes the sum of the payo¤s of all players in N . Henceforth, when referring to a cooperative outcome of the game, we allude to one of its PE. In our setting, as shown in the following Lemma, a PE strategy pro…le for every coalition S N (included the grand coalition) turns out to be always symmetric. P Lemma 1 For all S N; xe 2 arg maxxS 2XS i2S ui (xS ; xN nS ) implies xei = xej for all i,j 2 S and for all xN nS 2 XN nS : 4
xeS
Proof. Suppose xei 6= xej for some i; j 2 S: By symmetry we can derive from a new vector x0S by permuting the strategies of players i and j such that X X 0 ui (xS ; xN nS ) = ui (xeS ; xN nS ) (4) i2S
i2S
and hence, by the strict quasiconcavity of all ui (x); for all 2 (0; 1) we have that: X X 0 ui ( xS + (1 )xeS ; xN nS ) > ui (xeS ; xN nS ): (5) i2S
i2S
0
Since, by the convexity of X; the strategy vector xS + (1 )xeS 2 XS ; we obtain a contradiction. An important implication of Lemma 1 is that when the players of a given coalition S N deviate opportunistically by a given pro…le of strategy x 2 X n , they will all play the same maximizing strategy.
3
Main Results
In this section we present the main results of the paper. Let us …rst de…ne both a Symmetric Social Norm (SSN) and a Kinked Social Norm (KSN) of behaviour in the setting introduced above (see …gures 1-3 for SSN and KSN in a two-player case). De…nition 2 (Symmetric Social Norm) We say that a Symmetric Social Norm (SSN) is active in the game GS ; if every coalition of players S N deviating from a given pro…le of strategies x 2 X n with a x0S 2 X s such that x0S 6= xS , expects the following response sN nS (x0S ) : X s 7! X n s by players j 2 N nS : sN nS (x0S ) = f 8j 2 N nS; xj 2 Xj xj = x0i g :
(6)
De…nition 3 (Kinked Social Norm) We say that a Kinked Social Norm (KSN) is active in the game GS ; if every coalition of players S N deviating from a given pro…le of strategies x 2 X n with a x0S 2 X s such that x0S 6= xS , expects the following response kN nS (x0S ) : X s 7! X n s by players j 2 N nS : kN nS (x0S ) = f 8j 2 N nS; xj 2 Xj xj
x0i g :
(7)
x0i g :
(8)
under positive spillovers(PS) and kN nS (x0S ) = f 8j 2 N nS; xj 2 Xj xj under negative spillovers(NS). Note that in both cases there is no presumption of rational behaviour behind these norms and the reactions of players may not correspond to their best reply maps. Finally, let us introduce a general de…nition of stability of a strategy pro…le in our game GS under an arbitrary social norm. 5
De…nition 4 A strategy pro…le x 2 X n is stable under the social norm X s 7! X n s if there exists no S N and x0S 2 X s such that ui (x0S ; uh (x0S ;
0 N nS (xS )) 0 N nS (xS ))
N nS
:
ui (x) 8i 2 S; uh (x) for some h 2 S:
>
ui (xe ) 8i 2 S:
(9) 0
By the property of sN nS (xS ) and Lemma 1 it must be that sj (xS ) = xi for every j 2 N nS and i 2 S so that 0
ui (x ) > ui (xe ) for all i 2 N , implying
X
ui (x0 ) >
i2N
X
ui (xe )
(10)
(11)
i2N
contradicting that xe is a PE outcome. To show that no other strategy pro…le in GS can be stable under SSN, suppose by contradiction that an arbitrary ine¢ cient pro…le x 2 X n is stable under SSN. Therefore, for every i 2 N it must be that 0 0 ui (x) ui (xS ; sN nS (xS )): (12) 0
Take now xS = xeS ; since xeS 2 X s is an admissible deviation for every S N . It follows that ui (x) ui (xeS ; sN nS (xeS )) (13) and then by SSN ui (xe )
ui (x)
6
(14)
for all i 2 N; thus implying
X
X
ui (x)
i2N
ui (xe )
(15)
i2N
contradicting the ine¢ ciency of x (we know by Lemma 1 that every e¢ cient pro…le must be symmetric). Although the above result is not surprising since we expect that full coordination among players yields a PE allocation. However, if we relax the strict quasiconcavity of players’payo¤ and let asymmetric PE to exist in GS , they would not be stable either under SSN. In fact, with such asymmetric allocations at least one of the players receive less than in a SPE allocation and therefore he has an incentive, under SSN, to switch to the SPE. The next result concerns the alternative social norm de…ned above. Proposition 6 When a Kinked Social Norm (KSN) is active on GS , the cooperative outcome (SPE) xe 2 X n is stable. Proof. By De…nition 2, a KSN active on GS implies kN nS (x0S ) x0S under x0S under NS, for every x0S 2 X s : When kN nS (x0S ) = x0S PS and kN nS (x0S ) we know that, by Proposition 1, that xe 2 X n is stable. We need to prove that the same holds when kN nS (x0S ) < x0S under NS and kN nS (x0S ) > x0S under PS. Assume …rst that positive spillovers (PS) hold in GS . Assume also by contradiction that the SPE xe 2 X n is not stable and there exists a S N and a x0S 2 X s such that 0
0
ui (xS ; kN nS (xS )) > ui (xe ) 8i 2 S:
(16)
0
Using PE, Lemma 1 and the fact that kj (xS ) < x0i for every j 2 N nS and i 2 S, we obtain 0
0
0
0
ui (xS ; xN nS ) > ui (xS ; kN nS (xS )) > ui (xe ) and then, by the symmetry of GS X X ui (x0 ) > ui (xe ) i2N
(17)
(18)
i2N
a contradiction.
4
The Classical Kinked Demand Revisited
To illustrate our main results, let us introduce a simple example with two symmetric …rms. The original idea of the kinked demand model (Robinson 1936, Sweezy 1939) was the following: when a …rm, say …rm 1, rises its price, it expects that other …rms (here …rm 2) rise their price comaparatively less (underreaction). Conversely, when …rm 1 lowers its price, it expects …rm 2 to reduce 7
its price even more (over-reaction). This conjecture yields the well known model of kinked demand (see …gure 4). Suppose now that the two …rms are charging the consumers the pair of prices (p1 ; p2 ), and that these prices are perfectly cooperative for the …rms, i.e., they maximize the sum of pro…ts of the two …rms. The ’kinked demand’norm of behaviour dictates the following: 0
0
0
0
If p0i
>
if p0i
pi for i = 1; 2, then kj (pi ) < pi , for j 6= i:
1 (pe1 ; pe2 ). It is well known that in a model of price competition the e¤ect of a rise in competitors’prices yields a 0 (positive spillover). Thus, if positive e¤ect on a …rm’s pro…t, that is, @@pji 1
0
p01 ; k2 (p1 ) >
1
(pe1 ; pe2 ) ;
it must be that 1
By symmetry,
1
0
p01 ; p2 0
p01 ; p2 = X
0
p01 ; k2 (p1 ) >
1
2
i
1
(pe1 ; pe2 ) :
0
p01 ; p2 , and then, 0
p01 ; p2 >
i=1;2
X
i
(pe1 ; pe2 ) ;
i=1;2
contradicting the e¢ ciency of the perfectly cooperative outcome. The same result obviously holds when it is …rm 2 to deviate. This implies that when all …rms expect a kinked demand response from the other …rms, that is, when kinked social norm becomes the established norm of behaviour for all …rms, no pro…table deviations are possible from the perfectly collusive outcome (monopoly pricing). Surprisingly, the model easily extends to the case in which the …rms set quantities instead of prices. The ’kinked demand’norm of behavior now dictates the following: 0
0
0
0
if qi0
>
if qi0
qi for i = 1; 2, then kj (qi ) > qi , for j 6= i;
qi , for j 6= i; 0
where qi0 indicates any quantity di¤erent from qi , and kj (qi ) the quantity set by the rival as a response. Again, it is well known that in in a model of quantity competition the e¤ect of a rise in competitors’quantities yields a negative e¤ect 0, just because, this lowers on a …rm’s pro…t (negative spillovers), that is, @@qji 8
the market price p (q1 ; q2 ). Hence, if …rm 1 pro…tably deviates from the pair of 0 strategies (q1 ; q2 ), that is, 1 q10 ; k2 (q1 ) > 1 (q1 ; q2 ), it follows that 1
0
q10 ; q2
1
0
q10 ; k2 (q1 ) >
0
1
(p1 ; p2 ) :
0
Then, by symmetry, 1 q10 ; q2 = 2 q10 ; q2 , and this contraddicts the e¢ ciency of the pair of strategies (q1 ; q2 ).
5
Graphics
Figure 1 - SSN in a two-player case
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Figure 2
Figure 3
Figure 4
References [1] Bicchieri, C. (1990), "Norms of Cooperation", Ethics, 100, pp.838-861. 10
[2] Brown, A. D. (1995), Organizational Culture. London: Pitman [3] Castelfranchi, C., M. Miceli (2002). "The mind and the future: The (negative) power of expectations" Theory & Psychology,12, pp.335-366. [4] Fehr, E., U. Fischbacher and S. Gachter (2002), "Strong Reciprocity, Human Cooperation, and the Enforcement of Social Norms", Human Nature, 2003 [5] Hall, R.L. and Hitch, C. J.(1939), "Price Theory and Business Behaviour", Oxford Economic Papers, 2, pp.12-45. [6] Hart and Mas Collel’s (2000, 2001a, 2001b, 2003a, 2003b) [7] Lewis, D. K. (1969). Convention. Cambridge, Harvard University Press. [8] Robinson, J. (1933), "Economics of Imperfect Competition", London, Macmillan. [9] Sweezy, P. M. (1939), "Demand under Conditions of Oligopoly", Journal of Political Economy, 47, pp.568-573.
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