Proceedings of the 48th Annual Meeting of the Association for Computational Linguistics, pages 698–709,Uppsala, Sweden, 11-16 July 2010. c©2010 Association for Computational Linguistics

A Game-Theoretic Model of Metaphorical Bargaining

Beata Beigman KlebanovKellogg School of Management

Northwestern Universitybeata@northwestern.edu

Eyal BeigmanWashington University in St. Louis

beigman@wustl.edu

Abstract

We present a game-theoretic model of bar-gaining over a metaphor in the context ofpolitical communication, find its equilib-rium, and use it to rationalize observedlinguistic behavior. We argue that gametheory is well suited for modeling dis-course as a dynamic resulting from a num-ber of conflicting pressures, and suggestapplications of interest to computationallinguists.

1 Introduction

A 13 Dec 1992 article in The Times starts thus:

The European train chugged out of the stationlast night; for most of the day it looked as if itmight be stalled there for some time. It managedto pull away at around 10:30 pm only after theSpanish prime minister, Felipe Gonzalez, forcedthe passengers in the first class carriages into alast minute whip round to sweeten the trip for theEuropean Community’s poor four: Spain, Portu-gal, Greece and Ireland.

The fat controller, Helmut Kohl, beamed withsatisfaction as the deal was done. The elegantly-suited Francois Mitterrand was equally satisfied.But nobody was as pleased as John Major, sta-tionmaster for the UK presidency, for whom theagreement marked a scarce high point in a bat-tered premiership.

The departure had actually been delayed byseven months by Danes on the line. Just whenthat problem was solved, there was the volu-ble outbreak, orchestrated by Spain, from thepoor four passengers demanding that they shouldtravel free and be given spending money, too.

The coupling of the carriages may not be reli-ably secure but the pan-European express is inmotion. That few seem to agree the destinationsuggests that future arguments are inevitable atevery set of points. Next stop: Copenhagen.

Apart from an entertaining read, the extendedmetaphor provides an elaborate conceptual cor-respondence between a familiar domain of trainjourneys and the unfolding process of European

integration. Carriages are likened to nation states;passengers to their peoples; treaties to stations;politicians to responsible rail company employees.

In a compact form, the metaphor gives expres-sion to both the small and the large scale of theprocess. It provides for the recent history: Den-mark’s failure to ratify the 1992 Maastricht treatyuntil opt-outs were negotiated later that year iscompared to dissenters sabotaging the journey bylaying on the tracks (Danes on the line); nego-tiations over the Cohesion Fund that would pro-vide less developed regions with financial aid tohelp them comply with convergence criteria arelikened to second class carriages with poor pas-sengers for whom the journey had to be subsi-dized. At a more general level, the European in-tegration is a purposeful movement towards somedestination according to a worked out plan, get-ting safely through negotiation and implementa-tion from one treaty to another, as a train movingon its rails through subsequent stations, with eachnation being separate yet tied with everyone else.Numerous inferences regarding speed, timetables,stations, passengers, different classes of tickets,temporary obstacles on the tracks, and so on canbe made by the reader based on the knowledge oftrain journeys, giving him or her a feeling of an en-hanced understanding1 of the highly complex pro-cess of European integration.

So apt was the metaphor that political fightswere waged over its details (Musolff, 2000). Wor-ries about destination were given an eloquent ex-pression by Margaret Thatcher (Sunday Times, 20Sept 1992):

She warned EC leaders to stop their endlessround of summits and take notice of their ownpeople. “There is a fear that the European trainwill thunder forward, laden with its customarycargo of gravy, towards a destination neitherwished for nor understood by electorates. Butthe train can be stopped,” she said.

1More on enhanced understanding in sections 3.2 and 4.2.

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The metaphor proved flexible enough for fur-ther elaboration. John Major, a Conservative PMof Britain, spoke on June 1st, 1994 about his vi-sion of the decision making at the EU level, say-ing that he had never believed that Europe mustact as one on every issue, and advocating “a sensi-ble new approach, varying when it needs to, multi-track, multi-speed, multi-layered.” He attemptedto turn a largely negative Conservative take on theEuropean train (see Thatcher above) into a tenablepositive vision — each nation-carriage is now pre-sumably a rather autonomous entity, waiting on aside track for the right locomotive, in a huge yetsmoothly operating railroad system.

Major’s political opponents offered theircounter-frames. In both cases, the imagery ofa large transportation system was taken up, yetturned around to suggest that “multi, for every-one” amounts to Britain being in “the slow lane,”and a different image was suggested that makesthe negative evaluation of Britain’s opt-outsmore poignant — a football metaphor, whererelegation to the second division is a sign of aweak performance, and a school metaphor, whereBritain is portrayed as an under-achiever:

John Cunningham, Labour He has admitted that his Go-vernment would let Britain fall behind in Europe. Heis apparently willing to offer voluntary relegation to thesecond division in Europe, and he isn’t even prepared toput up a fight. I believe that in any two-speed Europe,Britain must be up with those in the fast lane. ClearlyMr Major does not.

Paddy Ashdown, Liberal Democrat Are you really sayingthat the best that Britain can hope for under your leader-ship is ... the slow lane of a two-speed Europe? Mostpeople in this country will want to aim higher, and willreject your view of a ‘drop-out’ Britain.

The pro-European camp rallied around the“Britain in the slow lane” version as a criticalstance towards the government’s European policy.Of the alternative metaphors, the school metaphorhas some traction in the Euro discourse, where theEuropean (mainly German) financial officers arecompared to school authorities, and governmentsstruggling to meet the strict convergence criteria toenter the Euro are compared to pupils that barelymake the grade with Britain as a ‘drop-out’ whogave up even trying (Musolff, 2000).

The fact that European policy is being commu-nicated and negotiated via a metaphor is not sur-prising; after all, “there is always someone willingto help us think by providing us with a metaphor

that accords with HIS views.”2 From the point ofview of the dynamics of political discourse, thepuzzle is rather the apparent tendency of politi-cians to be compelled by the rival’s metaphori-cal framework. Thatcher tries to turn the trainmetaphor used by the pro-EU camp around. Yet,assuming metaphors are matters of choice, whyshould Thatcher feel constrained by her rival’schoice, why doesn’t she ignore it and merely sug-gest a new metaphor of her own design? As theevidence above suggests, this is not Thatcher’sidiosyncrasy, as Major and his rivals acted simi-larly. Can this dynamic be explained?

In this article, we use the explanatory frame-work of game theory, seeking to rationalize the ob-served behavior by designing a game that wouldproduce, at equilibrium, the observed dynamics.Specifically, we formalize the notion that the priceof “locking” the public into a metaphorical frameof reference is that a politician is coerced into stay-ing within the metaphor as well, even if he or sheis at the receiving end of a rival’s rhetorical move.

Since the use of game theory is not common incomputational linguistics, we first explain its mainattributes, justify our decision to make use of it,and draw connections to research questions thatcan benefit from its application (section 2). Next,we design the game of bargaining over a metaphor,and find its equilibrium (section 3), followed by adiscussion (section 4).

2 Game-Theoretic models

The basic construct is that of a game, that is,a model of participants in an interaction (called“players”), their goals (or “utilities”) and allow-able moves. Different moves yield different util-ities for a player; it is assumed that each playerwould pick a strategy that maximizes her utility.The observable is the actual sequence of moves;importantly, these are assumed to be the optimaloutcome (an equilibrium) of the relevant game. Apopular notion of equilibrium is Nash equilibrium(Nash, 1950). For extensive form games (the typeemployed in this paper), the notion of subgameperfect equilibirum is typically used, denoting aNash equilibrium that would remain such if theplayers start from any stage of the evolving game(Selten (1975; 1965)).

The task of a game theorist is to reverse-engineer the model for which the observed se-

2Capitalization in the original, Bolinger (1980, p. 146).

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quence of actions is an equilibrium. The resultingmodel is thereby able to rationalize the observedbehavior as a naturally emerging dynamics be-tween agents maximizing certain utility functions.In economics, game-theoretic models are used toexplain price change, organization of production,and market failures (Mas-Colell et al., 1995; vonNeumann and Morgenstern, 1944); in biology —the operation of natural selection processes (Ax-elrod and Hamilton, 1981; Maynard Smith andPrice, 1973); in social sciences — political institu-tions, collective action, and conflict (Greif, 2006;Schelling, 1997; North, 1990). In recent appli-cations in linguistics, pragmatic phenoma such asimplicatures are rendered as an equilibrium out-come of a communication game (Jager and Ebert,2008; van Rooij, 2008; Ross, 2007; van Rooij andSchulz, 2004; Parikh, 2001; Glazer and Rubin-stein, 2001; Dekker and van Rooy, 2000).

Computing equilibria is simple for some gamesand quite evolved for others. For example, com-puting the equilibrium of a zero-sum game is equi-valent to LP optimization (Luce and Raiffa, 1957);an equilibrium of general bimatrix games can befound using a pivoting algorithm (von Stengel,2007; Lemke and Howson, 1964). Interestingconnections have been pointed out between gametheory and machine learning: Freund and Schapire(1996) present both online learning and boostingas a repeated zero-sum game; Shalev-Shwartz andSinger (2006) show similarly that loss minimiza-tion in online learning is akin to an equilibriumpath in a repeated game.

While game theoretic models are not much uti-lized in computational linguistics, they are quiteattractive to tackle some of the problems com-putational linguists are interested in. For exam-ple, generation of referring expressions (Paraboniet al., 2007; Gardent et al., 2004; Siddharthanand Copestake, 2004; Dale and Reiter, 1995) canbe rendered as a communication game with util-ity functions that reflect pressures to use shorterexpressions while avoiding excessive ambiguity(Clark and Parikh, 2007), with corpora anno-tated for entity mentions informing the designof a model. Generally, computational linguis-tics research produces algorithms to detect enti-ties of various kinds, be it topics, named entities,metaphors, moves in a multi-party conversations,or syntactic constructions in large corpora; suchprimary data can be used to trace developments

not only in chronological terms (Gruhl et al., 2004;Allan, 2002), but in strategic terms, i.e. in termsthat reflect agendas of the actors, such as politicalagendas in legislatures (Quinn et al., 2006) or ac-tivist forums (Greene and Resnik, 2009), researchagendas in group meetings (Morgan et al., 2001),or social agendas in speed-dates (Jurafsky et al.,2009). Game theoretical models are well suitedfor modeling dynamics that emerge under multi-ple, possibly conflicting constraints, as we exem-plify in this article.

3 The model

We extend Rubinstein (1982) model of negotia-tion through offers and counter-offers between twoplayers with a public benefit constraint.

The model consists of (1) two players repre-senting the opposing sides, (2) a set of framesX⊂Rn compact and convex, (3) preference re-lations described by continuous utility func-tions U1, U2:X→R+, (4) a sequence of framesX0⊂X1 . . .⊂2X that can be suggested to the pub-lic, and (5) a sequence of public preferences overframes inXt for t=0, 1, 2, . . . described by a publicutility function Up

t .The game proceeds as follows. Initially the

frame is F0=X . In odd rounds player 1 appeals tothe public with a frame A1

t∈Xt|Ft, Xt|Ft

={A∈Xt :A⊂Ft}, player 2 counters with a frame A2

t∈Xt|Ft.

The public chooses one of the frames based onUp

t (Ait) with ties broken in 1’s favor. The ac-

cepted frame becomes the current frame for thenext round Ft+1. In even rounds the parts of play-ers 1 and 2 are reversed.

A finite sequence F0, . . . , Ft−1 gives the his-tory of the bargaining process up to t. Astrategy σi of player i is a function specify-ing for any history h={F0, . . . , Ft−1} the moveplayer i makes at time t, namely the frame Ai

t

she chooses to address the public. A sequenceF0, F1, F2, F3, . . . describes a path the bargainingprocess can take, leading to an outcome ∩∞t=0Ft.The players’ utility for an outcome is given byUi=limt→∞

∫Ft

Ui(x)dχFt for i=1, 2 where χFt isa probability measure on Ft. If ∩∞t=0Ft={x} theutility is the point utility of x otherwise it is theexpected utility on the intersection set.

3.1 Player utility

For a given issue under discussion, such as Eu-ropean integration process, we order the possible

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states of the world along a single dimension thatspans the policy variations proposed by the diffe-rent players (politicians). Politics of a single issueare routinely modeled as lying on a single dimen-sion.3 In the British context, various configura-tions of the unfolding European reality are situatedalong the line between high degree of integrationand complete separatism; Liberal Democrats arethe most pro-European party, while United King-dom Independence Party are at the far-right end ofthe scale, preferring British withdrawal from theEU. The two major parties, Labour and Conserva-tives (Tories), prefer intermediate left-leaning andright-leaning positions, respectively. A schematicdescription is shown in figure 1.

!"#!$#

%"#

%$#

!"#!$#

%"#

%$#

LibDem! Labour! Tories! UKIP!

!"#$%&'()"*+,*-+.$*'*

#&'+"*/)0&"$12*

… that is unfolding

too fast

… but it is possible to

regulate the speed

… in which case we’ll go

slower than others

!"# !$#

%"#

%$#

Figure 1: Preferences on pro-anti Europe axis.

The utilities of the different players can in thiscase be described as continuous single-peakedfunctions over an interval.4 Thus X=[0, 1], andthe utility functions Ui(x)=φ(||x− vi||) for vi∈Xwhere φ is a monotonically strictly decreasingfunction and || || is Euclidean distance.

3.2 Public utilityWe note the difference between two types of util-ities: The utility of the players is over outcomes,the utility of the public is over sets of outcomes(frames). The latter does not represent a utility thepublic has for one outcome or another, but rather autility it has for an enhanced understanding. Thus,the public’s utility from a frame is a function ofthe information content of the proposed frame re-lative to the current frame, i.e. the relative en-tropy of the two sets.5 Formally, if the accepted

3Indeed, Poole and Rosenthal (1997) argue that no morethan two dimensions are needed to account for voting patternson all issues in the US Congress.

4Single-peakedness is a common assumption in positionmodeling in political science (Downs, 1957).

5The notion that new beliefs are refinements of existingones is current in contemporary theorizing about formationand change of beliefs, evaluations, and preferences. An up-date based on the latest available information is consistentwith memory-based theories; in our model, in the equilib-rium, the current frame contains information about the path-so-far, thus early stages of the bargaining processes are insome sense integrated into the current frame, compatible withthe rival, online model of belief formation. See Druckmanand Luria (2000) for a review of the relevant literature.

frame at time t is Ft then for any Borel set A⊂Ft

the public utility for A is Upt (A)=Π(Entt(A))

where Entt(A)=−µt(A) log µt(A) for a continu-ous probability measure µt on Ft and Π is a con-tinuous, monotone ascending function; for A 6⊂Ft,Up

t (A)=0. We take µt to be the relative length ofthe segment µt(A)= |A|

|Ft| , hence the entropy maxi-

mizing subsegments are of length |Ft|2 .

3.3 Game dynamics

At every point in the game, a certain set of thestates-of-affairs is being deemed sufficiently pro-bable by the public to require consideration. Sup-pose that initially any state of affairs within the in-terval [0, 1] is assigned a uniform probability andthus merits public attention. Each in her turn, theplayers propose to the public to concentrate ona subset of the currently considered states of af-fairs, arguing that those are the likelier ones to ob-tain, hence merit further attention. The metaphorused to deliver the proposal describes the newlyproposed subset in a way that makes those states-of-affairs that are in it aligned with the metaphor,whereas all other states are left out of the proposedmetaphorical frame. As the game proceeds, thepublic attention is concentrated on successivelysmaller sets of eventualities, and these are givena more and more detailed metaphoric description,providing the educational gratification of increa-singly knowing better and better what is going on.At each step, each player strives to provide maxi-mum public gratification while leading the publicto focus on the frame (i.e. subset of states of af-fairs) that best meets the player’s preferences.6

Figure 2 sketches the frame negotiation throughtrain metaphor, from some point in time when thegeneral train metaphor got established, throughThatcher’s flashing out the issue of excessivespeed and unclear direction, Major’s multi-trackcorrective, and reply of his opponents on the left.The final frame has all those states of affairs thatfit the extended metaphor – everyone is actingwithin the same broad system of rules, with Britainand perhaps others sometimes wanting to negoti-ate special, more gradual procedures, which wouldleave Britain less tightly integrated into the com-

6We note that in our model every utterance has an impacton the public for which the player bears the consequences andis therefore a (costly) strategic move in the game. This is dif-ferent from models of cheap talk such as Aumann (1990),Lewis (1969) where communication is devoid of strategicmoves and is used primarily as a coordination device.

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munity than some other European partners.

Integration is likea train journey…

… that isunfolding too fast

… but it is possible toregulate the speed

… in which case we’ll goslower than others

Figure 2: Bargaining over train metaphor.

3.4 The equilibriumA pair of strategies (σ1, σ2) is a Nash equilibriumif there is no deviation strategy σ such that (σ, σ2)leads to an outcome with higher utility for player 1than outcome of (σ1, σ2) and the same for player2. A subgame are all the possible moves followinga history h={F0, . . . , Ft}, in our case it is equi-valent to a game with an initial frame Ft and thecorresponding utilities. A sub-strategy is that partof the original strategy that is a strategy on thesubgame. A pair of strategies is a subgame per-fect equilibrium if, for any subgame, their sub-strategies are a Nash equilibrium.

Theorem 1 In the frame bargaining game withsingle-peaked preferences

1. There exists a canonical subgame perfectequilibrium path F0, F1, F2, . . . such that∩∞t=0Ft={x}.

2. For any subgame perfect equilibrium pathF ′

0, F′1, F

′2, . . . there exists T such that

∩∞t=0F′t=∩T

t=0Ft.

The theorem states that the outcome of the bar-gaining will always be a frame on the canoni-cal path. The rivals would suggest more specificframes either until convergence or until a situationwhere any further specification would produce aframe that “misses their point,” so-to-speak, by re-moving too much of the favorable outcome spacefor both players. Figure 3 shows a situation whereparties could decide to stall on the current frame:If player 1 has to choose between retaining F0, orplaying F1 which would result in the rival’s play-ing F2, player 1 might choose to remain in F0 ifthe utility of any outcome of the subgame startingfrom F2 is lower than that of F0, as long as player1 believes that player 2 would reason similarly.

F0

F2

F1

Player 1 Player 2

!"# !$#

Figure 3: Stalled bargaining.

The idea of the proof is to construct a pair ofstrategies where each side attempts to pull the pub-licly accepted frame in the direction of its peakutility point. We show, assuming the peak of thefirst mover is to the left of peak of the second, thatany deviation of the first mover would enable thesecond to shift the public frame more to the right,to an outcome of lower utility to the first mover.The full details of the proof of part 1 are given inthe appendix; part 2 is proved in an accompanyingtechnical report.

The equilibrium exhibits the following prop-erties: (a) a first mover’s advantage — for anyplayer, the outcome would be closer to her peakpoint if she moves first than if she moves second;(b) a centrist’s advantage — if a player moves firstand her peak is closer to the middle of the initialframe, she can derive a higher utility from the out-come than if her peak were further from the mid-dle. Please see appendix for justifications.

4 Discussion

4.1 Political communication

This article studies some properties of frame bar-gaining through metaphor in political communi-cation, where rival politicians choose how to ela-borate the current metaphor to educate the pub-lic about the ongoing situation in a way most con-sistent with their political preferences. Modelingthe public preferences as highest relative entropysubset of possible states-of-affairs, we show thatstrategic choices by the politicians lead to a sub-game perfect equilibrium where the less politicallyextreme player who moves first is at an advantage.

In a democracy, such player would typically bethe government, as the bulk of voters do not bydefinition vote for extreme views, and since thegovernment is the agent that brings about changesin the current states of affairs, and is thus the firstand most prepared to explain them to the public.Indeed, Entman’s model of frame activation in po-litical discourse is hierarchical, with the govern-

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ment (administration) being the topmost frame-activator, and opposition and media elites typi-cally reacting to the administration’s frame (Ent-man, 2003).

4.2 Metaphor in political communication

The role of metaphor in communication has longbeen a subject of interest, with views ranging froman ornament that beautifies the argument in theancient rhetorical traditions, to the contemporaryviews of conceptual metaphor as permeating everyaspect of life (Lakoff and Johnson, 1980).

In political communication specifically,metaphor has long been known as a framingdevice. Framing can be defined as “selectingand highlighting some facets of events or issues,and making connections among them in order topromote a particular interpretation, evaluation,or solution” (Entman, 2003). Metaphors arenotorious for allowing subliminal framing, wherethe metaphor seems so natural that the aspectsof the phenomenon in question that do not alignwith the metaphor are seamlessly concealed.For example, WAR AS A COMPETITIVE GAME

metaphor emphasizes the glory of winning and theshame of defeat, but hides the death-and-sufferingaspect of the war, which makes sports metaphorsa strategic choice when wishing to arouse apro-war sentiment in the audience (Lakoff, 1991).Such subliminal framing can often be effectivelycontested by merely exposing the frame.

Our examples show a different use of metaphor.Far from being subliminal or covert, the details ofthe metaphor, its implications, and the evaluationpromoted by any given version are an importanttool in the public discussion of a complex politi-cal issue. The function of metaphorical framinghere resembles a pedagogical one, where render-ing an abstract theory in physics (such as electri-city) in concrete commonsensical terms (such aswater flow) is an effective strategy to enhance thestudents’ understanding of the former (Gentnerand Gentner, 1983). The measure of success for agiven version of the frame is its ability to sway thepublic in the evaluative direction envisioned by theauthor by providing sufficient educational benefit,so-to-speak, that is, convincingly rendering a goodportion of a complex reality in accessible terms.

Once a frame is found that provides extensiveeducation benefit, such as the EUROPEAN INTE-GRATION AS TRAIN JOURNEY above, a politi-

cian’s attempt to debunk a metaphor as inappropri-ate risk public antagonism, as this would be akinto taking the benefit of enhanced understandingaway. Thus, rather than contesting the validity ofthe metaphoric frame, politicians strive to find away to turn the metaphor around, i.e. accept thegeneral framework, but focus on a previously un-explored aspect that would lead to a different eva-luative tilt. Our results show that being the firstto use an effective metaphor that manages to lockthe public in its framework is a strategic advantageas the need to communicate with the same publicwould compel the rival to take up the metaphorof your choice. To our knowledge, this is the firstexplanation of the use of extended metaphor in po-litical communication on a complex issue in termsof the agendas of the rival parties and the chang-ing disposition of the public being addressed. Itis an open question whether similar “locking in”of the public can be attained by non-metaphoricalmeans, and whether the ensuing dynamics wouldbe similar.

4.3 Social dynamics

This article contributes to the growing literature onmodeling social linguistic behavior, like debates(Somasundaran and Wiebe, 2009), dating (Juraf-sky et al., 2009; Ranganath et al., 2009), colla-borative authoring and editing in wikis (Leuf andCunningham, 2001) such as Wikipedia (Vuong etal., 2008; Kittur et al., 2007; Viegas et al., 2004).The latter literature in particular sees the social ac-tivity as an unfolding process, for example, detec-ting the onset and resolution of a controversy overthe content of a Wikipedia article through track-ing article talk7 and deletion-and-reversion pat-terns. Somewhat similarly to the metaphor debatediscussed in this article, Viegas et al. (2004) notefirst-mover advantage in Wikipedia authoring, thatis, the first version gives the tone for the subse-quent edits and has its parts survive for relativelymany editing cycles. Finding out how the ini-tial contribution constrains and guides subsequentedits of the content of a Wikipedia article and whatkind of argumentative strategies are employed inpersuading others to retain one’s contribution is aninteresting direction for future research.

A number of recent studies of the linguistic as-pects of social processes are construed as if the

7a page separate from the main article that is devoted tothe discussion of the edits

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events are taking place all-at-once — there is nodifferentiation between early and later stages of adebate in Somasundaran and Wiebe (2009) or ini-tial and subsequent speed-dates for the same sub-ject in Jurafsky et al. (2009). Yet adopting a dy-namic perspective stands to reason in such cases.

For example, Somasundaran and Wiebe (2009)built a system for recognizing stance in an onlinedebate (such as pro-iPhone or pro-Blackberry onhttp://www.covinceme.net). They noticed that thetask was complicated by concessions — acknow-ledgments of some virtues of the competitor be-fore stating own preference. This is quite possi-bly an instance of debate dynamics whereby as thedebate evolves certain common ground emergesbetween the sides and the focus of the debatechanges from the initial stage of elucidating whichfeatures are better in which product to a stagewhere the “facts” are settled and acknowledged byboth sides and the debate moves to evaluation ofthe relative importance of those features.

As another example, consider the constructionof statistical models of various emotional and per-sonality traits based on a corpus of speed datessuch as Jurafsky et al. (2009). Take the trait ofintelligence. In their experiment with speed-dates,Fisman et al. (2006) found that males tend to dis-prefer females they perceive as more intelligent orambitious than themselves. Consequently, an in-telligent female might choose to act less intelligentin later rounds of speed dating if she has not so farmet a sufficiently intelligent male, assuming sheprefers a less-intelligent male to no match at all.

Better sensitivity to the dynamics of social pro-cesses underlying the observed linguistic commu-nication will we believe result in increased inte-rest in game-theoretic models, as these are espe-cially well suited to handle cases where the sideshave certain goals and adapt their moves based onthe current situations, the other side’s move, andpossibly other considerations, such as the need toaddress effectively a wider audience, beyond thespecific interlocutors. A game theoretic explana-tion advances the understanding of the process be-ing modeled, and hence of the applicability, andthe potential adaptation, of statistical models de-veloped on a certain dataset to situations that dif-fer somewhat from the original data: For exam-ple, a corpus with more rounds of speed-datesper participant might suddenly make females seemsmarter, or a debate with a longer history would

feature more, and perhaps more elaborate, conces-sions.

5 Empirical challenges

We suggested that models of dynamics such asthe one presented in this article be built over datawhere entities of interest are clearly identified.This article is based on chapters 1 and 2 of thebook by Musolff (2000) which itself is informedby a corpus-linguistic analysis of metaphor in me-dia discourse in Britain and Germany. We nowdiscuss the state of affairs in empirical approachesto detecting metaphors.

5.1 Metaphors in NLP

Metaphors received increasing attention fromcomputational linguistics community in the lasttwo decades. The tasks that have been ad-dressed are explication of the reasoning behindthe metaphor (Barnden et al., 2002; Narayanan,1999; Hobbs, 1992); detection of conventionalmetaphors between two specific domains (Mason,2004); classification of words, phrases or sen-tences as metaphoric or non-metaphoric (Krishna-kumaran and Zhu, 2007; Birke and Sarkar, 2006;Gedigian et al., 2006; Fass, 1991).

We are not aware of research on automaticmethods specifically geared to recognition of ex-tended metaphors. Indeed, most computationalwork cited above concentrates on the detection ofa local incongruity due to a violation of selectionalrestrictions when the verb or one of its argumentsis used metaphorically (as in Protesters derailedthe conference). Extended metaphors are expectedto be difficult for such approaches, since many ofthe clauses are completely situated in the sourcedomain and hence no local incongruities exist (seeexamples on the first page of this article).

5.2 Data collection

Supervised approaches to metaphor detection needto rely on annotated data. While metaphors areubiquitous in language, an annotation project thatseeks to narrow the scope of relevant metaphorsdown to metaphors from a particular source do-main (such as train journeys) that describe a par-ticular target domain (such as European integra-tion) and are uttered by certain entities (such assenior UK politicians) face the problem of spar-sity of the relevant data in the larger discourse: Arandom sample of the size amenable to human an-

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notation is unlikely to capture in sufficient detailmaterial pertaining to the one metaphor of interest.

To increase the likelihood of finding mentionsof the source domain, a lexicon of words fromthe source domain can be used to select docu-ments (Hardie et al., 2007; Gedigian et al., 2006).Another approach is metaphor “harvesting” –hypothesizing that metaphors of interest would oc-cur in close proximity to lexical items representingthe target domain of the metaphor, such as the 4word window around the lemma Europe used inReining and Lonneker-Rodman (2007).

5.3 Data annotation

A further challenge is producing reliable anno-tations. Pragglejaz (2007) propose a methodo-logy for testing metaphoricity of a word in dis-course and report κ=0.56-0.70 agreement for agroup of six highly expert annotators. BeigmanKlebanov et al. (2008) report κ=0.66 for detec-ting paragraphs containing metaphors from thesource domains LOVE and VEHICLE with mul-tiple non-expert annotators, though other sourcedomains that often feature highly conventiona-lized metaphors (like structure or foundation fromBUILDLING domain) or are more abstract and dif-ficult to delimit (such as AUTHORITY) present amore challenging annotation task.

5.4 Measuring metaphors

A fully empirical basis for the kind of model pre-sented in this paper would also involve defininga metric on metaphors that would allow measu-ring the frame chosen by the given version of themetaphor relatively to other such frames – that is,quantifying which part of the “integration is a trainjourney” metaphor is covered by those states of af-fairs that also fit Thatcher’s critical rendition.

6 Conclusion

This article addressed a specific communicativesetting (rival politicians trying to “sell” to the pub-lic their versions of the unfolding realities and ne-cessary policies) and a specific linguistic tool (anextended metaphor), showing that the particularuse made of metaphor in such setting can be ratio-nalized based on the characteristics of the setting.

Various questions now arise. Given the cen-tral role played by the public gratification con-straint in our model, would conversational situa-tions without the need to persuade the public, such

as meetings of small groups of peers or phone con-versations between friends, tend less to the use ofextended metaphor? Conversely, does the use ofextended metaphor in other settings testify to theexistence of presumed onlookers who need to be“captured” in a particular version of reality — asin pedagogic or poetic context?

Considerations of the participants’ agendas andtheir impact on the ensuing dynamics of the ex-change would we believe lead to further interest ingame theoretic models when addressing complexsocial dynamics in situations like collaborativeauthoring, debates, or dating, and will augmentthe existing mostly statistical approaches with abroader picture of the relevant communication.

A Proof of Existence of a SubgamePerfect Equilibrium

For a segment [a, b] and a≤v1<v2≤b letU1(x)=φ(||x − v1||) and U2(x)=φ(||x − v2||)be utility functions with peaks v1 and v2, re-spectively. For a history h={F0, . . . , Ft} whereFt=[lt, rt], let σ∗1(h), player 1’s move, be de-fined as choosing Ft+1=[lt+1, rt+1] such that|Ft+1|= |Ft|

2 , and rt+1 is as close as possible tov1. σ∗2 sets lt+1 with respect to v2 in a symmet-ric fashion. Since Ft shrinks by half every round,limt→∞ lt=limt→∞ rt=x∗, converging to a point.We now show (σ∗1, σ

∗2) is an equilibrium by show-

ing that neither player has a profitable deviation.Notice that after the first round the subgame is

identical to the initial game with F1 replacing F0,and the roles of players reversed. Player 2 had noinfluence on the choice of F1, hence she has a pro-fitable deviation iff she has a profitable deviationon the continuation subgame where she is the firstmover. It thus suffices to show that the first mover(player 1) has no profitable deviations to establishthat (σ∗1, σ

∗2) is an equilibrium.

Since by definition σ∗2 always chooses an en-tropy maximizing segment, for player 1 to choosea non-entropy maximizing segment (more or lessthan half the length) amounts to yielding the roundto player 2, which is equivalent in terms of the re-sulting accepted frame to a situation where player1 chooses an entropy maximizing segment – thesame one chosen by player 2. Thus we need toconsider only deviations with entropy maximizingframes.

Step 1: Suppose σ′1 is a strategy of player 1 andlet F ′

0, F′1, F

′2, . . . be the sequence of frames on

705

the path corresponding to the pair (σ′1, σ∗2). Let

t0 be the first move deviating from the equilibriumpath, namely Ft0 6=F ′

t0 . We first show that Ft0−1

could not be (a) completely to the left of v1 or (b)completely to the right of v2. Suppose (a) holds.Then by definition rt0−2=rt0−1<v1, and, induc-tively, r0=rt0−1<v1; this contradicts r0=1 that fol-lows from F0=[0, 1]. Possibility (b) is similarlyrefuted. Therefore, the only two cases for Ft0−1

with respect to v1 are depicted in figure 4. Notethat this implies v1≤x∗≤v2.

!"# !$#

Case 2:

Case 1: Ft0!1

Ft0!1

rt0

Figure 4: Two cases of current frame location.

Step 2: In case 1, σ∗1 will choose frames of type[lt, v1] for any t≥t0, and σ∗2 will do the same onany history in the continuation game, hence theoutcome will eventually be v1. As this is player 1’speak utility point, she has no profitable deviation.

Step 3: In case 2, Ft0 is the leftmost entropymaximizing subsegment of Ft0−1 and the devia-tion F ′

t0 can only be a shift to the right namelyr′t0≥rt0 . If player 2 could choose [v2, rt0+1] givenrt0 , she can still choose the same frame given r′t0 ,so the outcome would be v2 and F ′

t0 was not pro-fitable. If player 2 could not choose [v2, rt0+1]given rt0 , implying that x∗<v2, but as a result ofthe deviation can now choose [v2, r

′t0+1], imply-

ing that the outcome would be v2, clearly player1 has not benefited from the deviation since U1

is descending right of v1. If player 2 still cannotchoose [v2, r

′t0+1] after the deviation, she would

choose the rightmost entropy maximizing segmentwith l′t0+1≥lt0+1. If this still allows player 1 todo [l′t0+2, v1] and hence to lead to v1 as the out-come, it was possible in [lt0+2, v1] as well, so noprofit is gained by having deviated. Otherwise,r′t0+2≥rt0+2.

Step 3 can be repeated ad infinitum to showthat r′t≥rt unless for some history h the de-viation enables σ2(h)=[v2, r

′t]. In the former

case we get limt→∞ r′t=x′≥x∗=limt→∞ rt where∩∞t=1F

′t={x′}. Since r′t and rt are to the right

of v1 and U1 is descending right of v1 it fol-lows that U1(x∗)≥U1(x′). In the latter casex′≥v2. Since Ft is never strictly to the right of v2,

x∗=limt→∞ lt≤v2≤x′, therefore U1(x∗)≥U1(x′).In either case the deviation σ′1 cannot result in abetter outcome for player 1. This finishes the proofthat (σ∗1, σ

∗2) is a Nash equilibrium.

Notice that (σ∗1, σ∗2) prescribe sub-strategies on

any subgame that are themselves Nash equilibriafor the subgames, hence (σ∗1, σ

∗2) is a subgame per-

fect equilibrium 2

First Mover’s Advantage: The proof of step3 shows that having the left boundary of the cur-rent frame further to the right cannot yield a bet-ter outcome for player 1. Yet, if player 1’s firstturn comes after that of player 2, she will startwith a current frame with the left boundary furtherto the right than the initial frame before player 2moved, since moving the left boundary is player2’s equilibrium strategy. Hence a player wouldnever achieve a better outcome starting second ifboth players are playing the canonical strategy.

Centrist’s Advantage: Let M be the middle ofF0. Consider a more extreme version of player 1— player 1#. Suppose w.l.g. v#

1 <v1≤M . In casev#1 <v1<v2, for all utilities u of the outcome of

dynamics vs player 2, if player 1# could attain u,player 1 could attain u or more; the reverse is nottrue, for example when |v#

1 − lt|< |Ft|2 ≤|v1 − lt|

and player 1 (or 1#) is moving first. In casev2<v#

1 <v1, if player 1 (or 1#) moves first, sheis able to force her peak point as the outcome. Ifv#1 <v2<v1, player 1 can force v1 as the outcome,

whereas player 1# would not necessarily be ableto force v#

1 , as player 2 would pull the outcometowards v2. Hence a first moving centrist is neverworse off, and often better off, than a first movingextremist.

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