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Biform Games

Biform Games
Biform Games

Biform Games:

Additional Online Material?

Adam Brandenburger?Harborne Stuart?

July2006

These appendices supplement Brandenburger and Stuart[1,2006],[2,2006](“Biform Games”). Appendix C uses the e?ciency results(Propositions5.1and5.2)in the main text to frame some of the ideas in the corporate-strategy literature.Appendix D contains further comments on the e?ciency results.

Appendix C:A Connection to Corporate Strategy

Corporate(multibusiness)strategy is the study of when‘the whole is worth more than the sum of the parts.’Alternatively put,corporate strategy is about the opportunities that arise because individual entities(divisions of a corporation,potential alliance partners,etc.)may each be maximizing the value they can capture,but appropriate joint action by the entities could capture more value in total.

The literature usually distinguishes two kinds of corporate strategy.The“market power”route involves joint action that leads to capturing a greater slice of the existing pie.A good example is when di?erent divisions(or businesses)engage in joint purchasing of an input,and thereby negotiate a lower unit cost.The“e?ciency”route involves joint action that leads to creating a larger pie, and capturing at least a part of the increase.The focus of the literature is on the second kind of corporate strategy,and we follow that here.

Our observation is that the e?ciency results in the main text(Section5)give us a way to classify di?erent kinds of e?ciency-oriented corporate strategies.Think of a corporate strategy as a way of addressing an ine?ciency.Then,by the results in Section5,we can then break this down into addressing a failure of Adding Up,or No Externalities,or No Coordination.(Of course,a given strategy could address more than one failure.)Here is a sketch of how this scheme works.

?Please see the acknowledgements in the main paper.o n lin e-b g-07-02-06

?Address:Stern School of Business,New York University,New York,NY10012,abranden@https://www.wendangku.net/doc/a817945946.html,, https://www.wendangku.net/doc/a817945946.html,/~abranden

?Address:Graduate School of Business,Columbia University,3022Broadway,New York,NY10027, hws7@https://www.wendangku.net/doc/a817945946.html,,https://www.wendangku.net/doc/a817945946.html,/~hws7

a.No Adding Up A big question in corporate strategy is that of the appropriate vertical scope of the?rm.Within this topic,the Holdup problem has,of course,been central(e.g.,Collis and Montgomery[3,1998,pp.108-111]).Here is a simple biform model of Holdup.There are two players—call them the upstream?rm U and the downstream?rm D.If U makes an upfront investment of$4,then U and D can transact and together create$7of(gross)value.Calculate added values:The overall pie is$3,D creates$0on its own,U creates?$4on its own.The added value of D is thus$7,the added value of U is$3.In the Core,the$3pie will be divided so that D gets between$0and$7,and U gets between?$4and$3.(Of course,these calculations simply re?ect the fact that the game is a bilateral monopoly between U and D.)Will U make the e?cient decision to invest?Not necessarily—which is the Holdup problem.(In the biform game formalism, it will if and only if its con?dence index is greater than4/7.)

Our analysis says that one way to understand the source of potential ine?ciency in Holdup is as a failure of AU.(NE and NC are clearly satis?ed.)Solutions to Holdup can be understood similarly. Vertical integration,bringing in a downstream competitor,etc.are ways of getting AU or closer to it.(For example,in its ideal form,vertical integration creates a one-player game,in which AU is automatically satis?ed.A second downstream competitor would reduce the gap between the sum of the added values and the pie,ensuring the upstream?rm more value.)But we are not suggesting any new solutions to Holdup,only pointing out that Holdup can be classi?ed in our scheme.

Another area of corporate strategy where failures of AU come up is in horizontal relationships, e.g.,in strategic alliances.Here,again,there may be ine?ciency—usually,underinvestment—if a party isn’t assured its added value.

b.Externalities Typical corporate-strategy issues that?t in here are the problem of transfer-ring knowledge across divisions of a corporation and managing a corporate brand.The externality is that one division could take an action to share its knowledge with another division that would increase the value created by the second division(together with its customers and suppliers).Or,it is that one division might make an investment in a common brand that is suitable for that division’s ‘image’but unsuitable for another division’s image—i.e.,lowers willingness-to-pay for the second di-vision’s product,and so decreases the value created by that division,its customers,and its suppliers. In both cases,NE might fail,and ine?ciency could result.Corporate-level knowledge and brand management are often discussed as appropriate strategies to overcome such potential ine?ciencies.

c.Coordination Ideas around scale and scope would?t here.Consider several divisions where each has to decide which of two suppliers to source from.This could be analyzed as a biform game,similar to Example5.2in the text,where the No and Yes labels correspond to choosing one or other supplier.The e?cient pro?le of choices would be when each division chooses the supplier corresponding to the Yes label.But each choosing the other supplier(the No label)would also be a Nash equilibrium.We would again have a coordination game,and a failure of NC.The gains from coordination would come from the usual reasons of scale and scope—opportunities to share in defraying the?xed costs of the Yes supplier,to speed this supplier’s movement down the learning

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curve,etc.But with a failure of NC,there is no guarantee that the divisions,acting on their own, will make the e?cient choice.Solutions to this kind of potential ine?ciency are discussed in the literature under precisely the heading of coordination of the multibusiness?rm(Collis-Montgomery [3,1998,p.156]).

This is obviously not meant to be an exhaustive discussion of corporate strategy.Its purpose is simply to suggest that the e?ciency framework of AU,NE,and NC may be a useful way to classify and unify some of the corporate-strategy literature.It does also suggest a‘problem-driven’approach to crafting corporate strategies:try to spot ine?ciencies in the organization(or across organizations)and then?nd ways of correcting them.

Appendix D:Comments on E?ciency Results

i.E?ciency and Rationality We start by establishing a further implication of the AU,NE,and NC conditions.1

Proposition D1Consider a biform game(S1,...,S n;V;α1,...,αn)satisfying AU,NE,and NC, and that for each s∈S,the game V(s)has a nonempty Core.Then each player has a(strongly) dominant strategy.

Proof.Fix a player i and two strategies r i,s i∈S i.Suppose that V(r i,s?i)(N)>V(s i,s?i)(N) for some s?i∈S?i.Then NC implies V(r i,r?i)(N)>V(s i,r?i)(N)for all r?i∈S?i.Applying NE yields

V(r i,r?i)(N)?V(r i,r?i)(N\{i})>V(s i,r?i)(N)?V(s i,r?i)(N\{i})

for all r?i∈S?i.By AU,strategy r i strongly dominates strategy s i.

The remaining case is that V(r i,s?i)(N)=V(s i,s?i)(N)for all s?i∈S?i.But then NE yields V(r i,s?i)(N)?V(r i,s?i)(N\{i})=V(s i,s?i)(N)?V(s i,s?i)(N\{i})

for all s?i∈S?i.By AU,strategy r i is payo?-equivalent to strategy s i for player i.Finiteness of the strategy sets then implies that each player has a strongly dominant strategy(up to repetition of payo?-equivalent strategies).

Since a pro?le of dominant strategies is a Nash equilibrium,we know by Proposition5.1in the text that it is e?cient.Indeed,we can put Propositions5.1and5.2in the text,and Proposition D1 above,together in the following way.In a noncooperative game,a player is rational if he chooses a strategy that maximizes his expected payo?,under some probability measure on(the product of)the strategy sets of the other players.In a biform game,each player still associates a single 1John Sutton prompted us to prove this result.

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number with each strategy pro?le(equal to a weighted average of the Core projection),and so we can de?ne the rational strategic choices of a player correspondingly.With this terminology,we have the following equivalence statement:

Assume AU,NE,and NC(and nonemptiness of the Core for each strategy pro?le).Then,a pro?le consists of rational strategies if and only if the pro?le is e?cient.

The proof is immediate.By Proposition E1,under AU,NE,and NC,each player has a strongly dominant strategy(or strategies).This is then the only rational strategy(or strategies)for that player,and we just said that a pro?le of dominant strategies is e?cient.Conversely,Proposition

5.2says that under AU and NE(even without NC),an e?cient pro?le is a Nash equilibrium.And

a Nash-equilibrium strategy is rational—when the player assigns probability one to the other players’strategies that make up the equilibrium.

Viewed this way,our results provide an answer to the question:“When is rationality equivalent to e?ciency?”When are these two basic concepts,one from game theory and the other from welfare theory,equivalent?Our answer is that a su?cient condition for this is that AU,NE,and NC hold.

ii.Weak No Coordination Next,we note a possible weakening of the NC condition,and a corresponding counterpart to Proposition D1(the proof mirrors the one above).

De?nition D1A biform game(S1,...,S n;V;α1,...,αn)is said to satisfy Weak No Coordina-tion(WNC)if for each i=1,...,n;r i,s i∈S i;and r?i,s?i∈S?i,

V(r i,r?i)(N)≥V(s i,r?i)(N)if and only if V(r i,s?i)(N)≥V(s i,s?i)(N).

Proposition D2Consider a biform game(S1,...,S n;V;α1,...,αn)satisfying AU,NE,and WNC, and that for each s∈S,the game V(s)has a nonempty Core.Then each player has a weakly dom-inant strategy.

https://www.wendangku.net/doc/a817945946.html,ck of Necessity.In Section5in the text,we mentioned the lack of necessity of the conditions of Propositions5.1and5.2.To see that the lack of necessity in Proposition D1above, consider the Prisoner’s Dilemma(Figure D1),viewed as a biform game satisfying AU.(To see the game this way,note that AU in a two-player game implies that the game is inessential,so that the unique Core allocation gives the players their individually rational payo?s.These are the payo?s shown.)Here,NC holds but NE fails.Yet each player has a dominant strategy.

Figure D1

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To see the lack of necessity in the equivalence statement in subsection i.above,consider Matching Pennies(Figure D2),viewed as a biform game satisfying AU.Here,as in Figure D1,NC holds but NE fails.Yet each strategy of either player is rational and all pro?les are e?

cient.

Figure D2

iv.Relationship to Makowski-Ostroy In Section5,we mentioned the di?erence between our de?nition of added value(marginal contribution)and the de?nition in Makowski-Ostroy([4,1994], [5,1995]).We talk about player i’s marginal contribution(de?ned in the standard game-theoretic way)in each of the cooperative games V(s)induced by di?erent?rst-stage strategy pro?les s∈S. Let us now cast the M-O de?nition in the biform model(recognizing that their model is di?erent). M-O would select a particular strategy s i0∈S i of player i,to be thought of as the strategy of“not participating in the game.”They would then say that player i receives his marginal contribution if

his payo?from choosing any strategy s i,when the other players choose strategies s?i,is equal to V(s)(N)?V(s i0,s?i)(N\{i}).That is,player i receives the di?erence between the value created when i is“in the game”(and playing s i)and the value created by the remaining players when i is“out of the game”(playing s i0).By contrast,we say that i receives his marginal contribution if the payo?is V(s)(N)?V(s)(N\{i}).Note that if NE is assumed,then V(s)(N\{i})=V(s i0,s?i)(N\{i}), so that the two de?nitions then coincide.Thus,seen from the perspective of our formalism,M-O e?ectively assume NE from the beginning,while we distinguish games that do or do not satisfy NE.2 Another di?erence between M-O and us is in the treatment of coordination.M-O use a condition (“No Complementarities”)which,in the formalism of the biform model,says:For each r,s∈S,

V(s)(N)?V(r)(N)≤

n

X i=1£V(s i,r?i)(N)?V(r)(N)¤.

In words,the change in the value of the game when the players switch from strategy pro?le r to strategy pro?le s is no more than the sum of the changes caused by the players’switching one at a time.It can be shown that this condition holds if and only if we have:For each i=1,...,n;

2As a consequence,M-O would classify some examples di?erently from us.Thus,we attributed the ine?ciency in Example5.1to a failure of NE.M-O would attribute it to?rm1’s receiving$1from negative advertising,di?erent from the marginal contribution of advertising,which M-O would calculate as?$1.(Take the strategy of“not participating in the game”for?rm1to be what we called the“status-quo”strategy.)We are grateful to Louis Makowski for much help on the di?erence between our formalisms.

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r i,s i∈S i;and r?i,s?i∈S?i,

V(r i,r?i)(N)?V(s i,r?i)(N)=V(r i,s?i)(N)?V(s i,s?i)(N).

By standard arguments,this latter condition holds if and only if the value of the game is additively separable as a function of the players’strategy choices.Our No Coordination condition(De?nition 5.2)is weaker than the M-O condition,and,in particular,does not impose additive separability.

6

References

[1]Brandenburger,A.,and H.W.Stuart,Jr.,“Biform Games,”2006.Forthcoming in Management

Science.

[2]Brandenburger,A.,and H.W.Stuart,Jr.,“Biform Games:Online Appendix,”2006.Forthcoming

in Management Science.

[3]Collis,D.,and C.Montgomery,Corporate Strategy:A Resource-Based Approach,McGraw-Hill,

1998.

[4]Makowski,L.,and J.Ostroy,“Is Giving People Their Marginal Products Good for Incentives?

Revisiting the Pigovian Tradition,”unpublished,Department of Economics,University of Cali-fornia at Davis,1994.

[5]Makowski,L.,and J.Ostroy,“Appropriation and E?ciency:A Revision of the First Theorem

of Welfare Economics,”American Economic Review,85,1995,808-827.

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(完整版)英语口语试题及答案

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