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<Paper uid="C96-1090">
  <Title>Issues in Communication Game</Title>
  <Section position="3" start_page="0" end_page="531" type="metho">
    <SectionTitle>
2 Communication Games
</SectionTitle>
    <Paragraph position="0"> Communication has been discussed in the gametheory literature. A signaling game consists of sender S's sending a message (or a signal) to receiver R and R's doing some action in response to that message. Here S knows something that R did not know before receiving the message. This is formulated by assuming that S belongs to some type, which S knows but R does not know at first.</Paragraph>
    <Paragraph position="1"> Let T be the set of the types, P be the probability distribution over T. Let M be the set of the messages and A be the set of R's possible actions. Finally, let Ux be the utility function for player X. Us(t,m,a) and UR(t,m,a) are real numbers for t E T, m 6 M and a E A. A signaling game with T = {h,t~}, M : {ml,m2} and A = {al, a2} is illustrated by a game tree as shown ~If /~ is sincere and unintentionally misunderstands, that is just a failure of sharing the same context with S.</Paragraph>
    <Paragraph position="2">  in Figure 1. Here the game proceeds downwards.</Paragraph>
    <Paragraph position="4"> The top branch is the nature's initial choice of S's type according to P, the middle layer is S's decision on which message to send, and finally the bottom layer is R's choice of her action. When R has just received ml (i = 1, 2), she does not know whether the game has been played through tl or t2.</Paragraph>
    <Paragraph position="5"> Let as and ~n be S's and R's strategies, 3 respectively. That is, as(mlt ) is the conditional probability of S's sending message m provided that she is of type t, and an(aim ) the conditional probability of R's doing action a provided that she has received m. The combination (~s, an) of strategies is an equilibrium 4 of a signaling game when as and a~ are the optimal responses to each other; that is, when az maximizes X's expected</Paragraph>
    <Paragraph position="7"> given ay, for both X = SAY = R and X = RAY=S.</Paragraph>
    <Paragraph position="8"> In typical applications of signMing game, T, M and A are not discrete sets as in the above example but connected subsets of real numbers, and S's preference for R's action is the same irrespective of her type. In this setting, S should send a costly message to get a large payoff. For instance, in job market signaling (Spence, 1973), a worker S signals her competence (type) to a potential employer R with the level of her education as the message, and R decides the amount of salary to offer to S. A competent worker will have high education and the employer will offer her a high salary. In mate selection (Zahavi, 1975), a deer S indicates its strength by the size of its antlers  cise, technical term.</Paragraph>
    <Paragraph position="9"> to potential mates R. A strong deer will grow extra large antlers to demonstrate its extra survival competence with this handicap.</Paragraph>
    <Paragraph position="10"> Cheap-talk game is another sort of communication game. It is a special case of signaling game where Us and UR do not depend on the message; that is, composing/sending and receiving/interpreting message are free of cost. In a cheap-talk game, S's preference for R's action must depend on her type for non-trivial communication to obtain, because otherwise S's message would give no information to R about her type.</Paragraph>
  </Section>
  <Section position="4" start_page="531" end_page="533" type="metho">
    <SectionTitle>
3 Meaning Game
</SectionTitle>
    <Paragraph position="0"> Now we want to formulate the notion of meaning game to capture nonnatural meaning in the restricted sense discussed in Section 1. Let C be the set of semantic contents and P the probability distribution over the linguistic reference to the semantic contents. That is, P(c) is the probability that S intends to communicate semantic content c to R. As before, M is the set of the messages.</Paragraph>
    <Paragraph position="1"> A meaning game addresses a turn of communication (cs,m,c~&gt;, which stands for a course of events where S, intending to communicate a semantic content Ks, sends a message m to R and R interprets m as meaning CR. CS = cn is a necessary condition for this turn of communication to be successful. It seems reasonable to assume that the success of communication is the only source of positive utility for any player.</Paragraph>
    <Paragraph position="2"> So a meaning game might be a sort of signaling game in which S's type stands for her intending to communicate some semantic content, and R's action is to infer some semantic content. That is, both T and A could be simply regarded as C.</Paragraph>
    <Paragraph position="3"> Strategies ors and an are defined accordingly.</Paragraph>
    <Paragraph position="4"> In a simple formulation, the utility function Ux of player X would thus be a real-valued function from C x M x C (the set of turns). It would be sensible to assume that Ux(cs,m, eR) &gt; 0 holds only if es = cn. Ux reflects the grammar of the language (which might be private to S or R to various degrees). The grammar evaluates the (corn= putational, among others) cost of using content-message pairs. The more costly are (cs, m I and (m, cR), the smaller is Uz(cs, m, cn). The notion of equilibria in a meaning game is naturally derived from that in a signaling game.</Paragraph>
    <Paragraph position="5"> If the players want something like common belief, 5 however, meaning games are not signaling games. This is because cs = cn is not a sufficient condition for the success of communication in that case. Ux should then depend on not just KS, m, and c~, but also the players' nested beliefs 5People have common belief of proposition p when they all believe p, thcy all believe that they M1 believe p, they all believe that they all believe that they all believe p, and so on, ad infinitum.</Paragraph>
    <Paragraph position="6">  about each other. We will come back to this issue in Section 4.</Paragraph>
    <Paragraph position="7"> Note also that the typical instances of meaning game in natural language communication is not like the typical applications of signaling game such as mentioned before, even if meaning games are special sort of signaling games. That is, meaning games in natural language would normally involve discrete sets of semantic contents and messages.</Paragraph>
    <Paragraph position="8"> Natural-language meaning games are not cheap-talk games, either, because we must take into consideration the costs of content-message pairs. It is not just the success of communication but also various other factors that account for the players' utility. S and R hence do not just want to maximize the probability of successful communication. To illustrate a meaning game and to demonstrate that meaning games are not cheap-talk games, let us consider the following discourse.</Paragraph>
    <Paragraph position="9">  (1) ul: Fred scolded Max.</Paragraph>
    <Paragraph position="10"> u2: He was angry with the man.</Paragraph>
    <Paragraph position="11">  The preferred interpretation of 'he' and 'the man' in u~ are Fred and Max, respectively, rather than the contrary. This preference is accounted for by the meaning game as shown in Figure 2. In this probability: /)1 &gt; P2  game, Fred and Max are semantic contents, and 'he' and 'the man' are messages. 6 We have omitted the nature's selection among the semantic contents. Also, the nodes with the same label are collapsed to one. S's choice goes downward and R's choice upward, without their initially knowing the other's choice. The complete bipartite connection between the contents and the messages means that either message can mean either content grammatically (without too much cost).</Paragraph>
    <Paragraph position="12"> -Pl and P2 are the prior probabilities of references to Fred and Max in u2, respectively. Since Fred was referred to by the subject and Max by the object in ul, Fred is considered more salient than Max in u2. This is captured by assuming P1 &gt; P2. U1 and /72 are the utility (negative 6Perhaps there are other semantic contents and  messages.</Paragraph>
    <Paragraph position="13"> cost) of using 'he' and 'the man,' respectively, r Utilities are basically assigned to content-message pairs, but sometimes it is possible to consider costs of messages irrespective of their contents. We assume U1 &gt; U~. to the effect that 'he' is less complex than 'the man' both phonologically and semantically; 'he' is not only shorter than 'the man' but also, more importantly, less meaningful in the sense that it lacks the connotation of being adult which 'the man' has.</Paragraph>
    <Paragraph position="14"> There are exactly two equilibria entailing 100% success of communication, as depicted in Figure 3 with their expected utilities ~1 and PS2 apart from the utility of success of communication, s P, &gt; P2  and/71 &gt; U2 imply ~1 -~2 = (P1 -/)2)(U1 -U2) 0. So the equilibrium in the left-hand side is preferable for both S and R, or Pareto superior.</Paragraph>
    <Paragraph position="15"> This explains the preference in (1). It is straight-forward to generalize this result for cases with more than two contents and messages: A more salient content should be referred to by a lighter message when the combinations between tile contents and the messages are complete. A general conjecture we might draw from this discussion is the following.</Paragraph>
    <Paragraph position="16"> (2) Natural-language meaning games are played at their Pareto-optimal equilibria.</Paragraph>
    <Paragraph position="17"> An equilibrium is Pareto optimal iff no other equilibrium is Pareto superior to it.</Paragraph>
    <Paragraph position="18"> Note that we have derived an essence of centering theory (Joshi and Weinstein, 1981; Kamcyama, 1986; Walker et al., 1994; Grosz et al., 1995). Centering theory is to explain anaphora in natural language. It considers list Cf(ui) of forward-looking centers, which are the semantic entities realize~ in ui, where ul is the i-th utte&gt; ance. The forward-looking centers of utterance u 7For the sake of simplicity, here we assume that Us and U~ arc equal. See Section 4 for discussion.</Paragraph>
    <Paragraph position="19"> SCommon belief about the communicated content is always obtained in both cases. So the current discussion does not depend on whether the success of communication is defined by cs = cR or common belief.</Paragraph>
    <Paragraph position="20"> 9A linguistic expression realizes a semantic content when the former directly rcfers to the latter or the situation described by the former involves the latter.  are ranked in Cf(u) according to their saliences. In English, this ranking is determined by grammatical functions of the expressions in the utterance, as below.</Paragraph>
    <Paragraph position="21"> subject &gt; direct object &gt; indirect object &gt; other complements &gt; adjuncts The highest-ranked element of Cf(u) is called the preferred center of U and written Cp(u).</Paragraph>
    <Paragraph position="22"> Backward-looking center Cb(ui) of utterance ui is the highest-ranked element of Cf(ui-1) that is realized in ui. Cb(u) is the entity which the discourse is most centrally concerned with at u. Centering theory stipulates the following rule.</Paragraph>
    <Paragraph position="23">  (3) If an element of Cf(ui_J is realized by a pro- null noun in ui, then so is Cb(u{).</Paragraph>
    <Paragraph position="24"> In (1), Cb(u2) : Fred because Cf(ul) : \[Fred, Max\], if either 'he' or 'the man' refers to Fred. Then rule (3) predicts that Fred cannot be realized by 'the man' if Max is realized by 'he' -- the same prediction that we derived above. Moreover, (3) itself is a special instance of our above observation that a more salient content should be referred to by a lighter message, provided that the backward-looking center is particularly salient. (3) is common in all the version of centering theory, but of course there are further details of the theory, which vary from one version to another. To derive all of them (which are right) in a unified manner requires further extensive study.</Paragraph>
  </Section>
  <Section position="5" start_page="533" end_page="533" type="metho">
    <SectionTitle>
4 Playing the Same Game
</SectionTitle>
    <Paragraph position="0"> We have so far assumed implicitly that S and R have common knowledge about (the rule of) the game (that is, P, Us and UR). This assumption will be justified as a practical approximation in typical applications of signaling games (and cheap-talk games). For instance, there may well be a body of roughly correct, stable common-sense knowledge about the correlation between the competence of workers and the degree of effort they make to have higher education, about how much an employer will offer to an employee with a certain competence, and so on.</Paragraph>
    <Paragraph position="1"> However, common knowledge on the game might be harder to obtain in natural-language meaning games, because the game lacks such stability of the typical signaling games as mentioned above. A natural-language meaning game is almost equivalent to the context of discourse, which changes dynamically as the discourse unfolds.</Paragraph>
    <Paragraph position="2"> In general, to figure out her own best strategy, S (R) attempts to infer R's (S's) strategy by simulating R's (S's) inference. If S and R do not have common knowledge about the game, this inference will constitute an infinite tree. 1deg For instance,  knowledge of C : {Cl, c2\] and M : {ml, m2} but not of their utility functions. The nodes labeled by c~ represent S when she wants to communicate c~, and those labeled by m~ represent R when she wants to interpret mi, for i = 1, 2. The inference by R when interpreting message mi is a similar tree rooted by mi.</Paragraph>
    <Paragraph position="3"> Although it is impossible to actually have common knowledge in general (Halpern and Moses, 1990), there are several motivations for the players to pretend to have common knowledge about the game. First, they can avoid the computational complexity in dealing with infinite trees such as above. Second, common belief on the game is a simple means to obtain common belief on the communicated content. Third, the best payoff is obtained when the players have common knowledge about the game, if their utility functions are equal. In fact, the utility functions are probably equal, because language use as a whole is a repeated game. That is, provided that communicating agents play the role of S and R half of the time each, they can maximize their expected utility by setting their utility functions to the average of their selfish utilities. Fortunately, this equalization is very stable, as long as the success of communication is the only source of positive utility for both the players.</Paragraph>
    <Paragraph position="4"> In communication games, common knowledge on which message S has sent should help the players converge on common belief on the game.</Paragraph>
    <Paragraph position="5"> That is, when the players have common knowledge that message m was sent, they may be able to detect errors in their embedded beliefs. In fact, an embedded belief turns out wrong if it implies ~rs(mlc ) = 0 for every c in the embedded context.</Paragraph>
    <Paragraph position="6"> This common knowledge about m may be even incorporated in the meaning game. That is, it may affect the cost of retrieving or composing various content-message pairs, thus biasing the scope of the game towards those content-message pairs closely associated with m. Contents and messages very difficult to envisage given m will be virtually excluded from the game. Once the game is defined, however, both players must take into consideration the entire maximal connected subgraph containing the content she wants to convey or the message she wants to interpret.</Paragraph>
  </Section>
  <Section position="6" start_page="533" end_page="535" type="metho">
    <SectionTitle>
5 Composite Game
</SectionTitle>
    <Paragraph position="0"> Natural-language communication is a composite game in two senses. First, as mentioned in the previous section, it is considered a repeated game, which is a sequence of smaller games. Second, each such smaller game is a compound game consisting of temporally overlapping meaning games.</Paragraph>
    <Paragraph position="1"> These facts introduce several complications into the communication game.</Paragraph>
    <Paragraph position="2">  In a repeated game, one stage may affect the subsequent stage. In natural-language communication, a meaning game can influence the next meaning game. For instance, if a semantic content c is referred to by a message with a low cost, then the probability of reference to c may increase as a sort of accommodation, 1~ because a reference by a lightweight message presupposes high prior probability of reference, as discussed in Section 3. For instance, a reference to Fred by 'he' will raise the salience of Fred.</Paragraph>
    <Paragraph position="3"> Another type of contextual effect shows up the following discourse.</Paragraph>
    <Paragraph position="4"> (4) ul: Fred scolded Max.</Paragraph>
    <Paragraph position="5"> u2: The man was angry with him.</Paragraph>
    <Paragraph position="6"> Here 'the man' and 'he' in u2 are more readily interpreted as Fred and Max, respectively, which violates (3) and hence our game-theoretic account. This preference is accounted for by the preference for parallelism concerning the combination of semantic content and grammatical function: In both ul and u~. Fred is realized by the subject NP and Max is realized by the object NP. This is the same sort of preference that is addressed by property-sharing constraint (Kameyama, 1986).</Paragraph>
    <Paragraph position="7"> This effect is attributed to the utility assignment as shown in Figure 5. That is, the utility U1 of associating t he proposition a ngry(Fred, Max) (that l~ed is angry with Max) with the sentence 'The man was angry with him' is greater than the utility (/2 of associating angry(Max,Fred) (the proposition that Max is angry with Fred) with the same 11Lewis (1979) discusses several types of accommodation for conversationM score, of which the most relevant here is accommodation for comparative salience: x becomes more salient than y when something is said which presupposes x to be more salient than y.</Paragraph>
    <Paragraph position="8">  sentence. This game might involve other possible associations such as that between angry(Max,Fred) and 'The man made him angry,' but as mentioned at the end of Section 4 contents and messages other than included in Figure 5 probably accompany great costs and hence may be neglected. In general, several meaning games are played possibly in parallel during linguistic communica~ tion using a compound expression. A turn of corn= munication with an utterance of 'the man was angry with him' consists of the sentence-level game mentioned above, the two noun phrase-level games -- one concerning the subject NP (shown in Figure 2) and the other the object NP of 'with' -and so on. A strategy of each player in such a compound game associated with a compound expression is a combination of her strategies for all such constituent games. Each player attempts to maximize the expected utility over the entire compound game, rather than for each constituent game.</Paragraph>
    <Paragraph position="9"> Different constituent games often interact. For instance, if the speaker chooses to say 'the man' for the subject NP, then the whole sentence can- null not be 'he was angry with the man.' So a global solution, which maximizes the utility from the entire game, may maximize the utility from some constituent games but not from others. In the above example, the global solution, which involves saying 'the man was angry with him' and interpreting it as angry(Fred,Max), maximizes the utility from the sentence-level game but not from the NP-level games. Incidentally, the players will gain greater utility if they use the combination of angry(Fred,Max) and 'he was angry with the man,' which is consistent with the optimal equilibrium of the NP-games. When 'the man was angry with him' is used despite the smaller default utility associated with it, Max will probably be assigned a greater salience than otherwise, which is again a sort of accommodation.</Paragraph>
    <Paragraph position="10"> Extralinguistic context enters sentence-level games and plays an important role in language use. For example, if it is known that Max never gets angry and that Fred is short-tempered, then both in (1) and (4) the second utterance will preferably be interpreted as meaning angry(Fred,Max). null</Paragraph>
  </Section>
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