Signals without teleology

“Signals” are a conceptual apparatus in many scientific disciplines. Biologists inquire about the evolution of signals, economists talk about the signaling function of purchases and prices, and philosophers discuss the conditions under which signals acquire meaning. However, little attention has been paid to what is a signal. This paper is an attempt to fill this gap with a definition of signal that avoids reference to form or purpose. Along the way we introduce novel notions of “information revealing” and “information concealing” moves in games. In the end, our account offers an alternative to teleological accounts


Introduction
"Clouds mean rain" "The word 'dog' means dog" Distinguishing between the two senses of the verb 'to mean' in the above sentences has occupied a central place in the philosophical discussion of language. The difference was captured by Peirce's distinction between indices Finally, the same set of problems emerges in the study of economic behavior. Since Spence's work [17] in the 1970's economists have come to recognize that many economic actions like setting prices or paying for celebrity endorsements might function to convey information from one party to another.
Here, too, a critical distinction arises between those actions that are taken for the purpose of conveying information and those that convey information only incidentally. This distinction separates those economic behaviors which are truly signals from those that have another economic motivation. The study of "signaling games" has become a significant enterprise in economic thought. Even more radically, there is some recent interest in so-called "selfsignaling" where an economic action -like donating to a charity -is taken to purportedly send a signal to oneself at a future time [4].
All the methods for drawing the distinction create complexities for those interested in determining whether a particular sign is an index or a signal.
One must determine the provenance of a sign by finding either the intentions that were present at its production or the evolutionary processes that resulted in its fixation. In some cases this may be possible, but in other situations it might be difficult. For instance, in studies of animal behavior scientists may not know the evolutionary history leading to the production of a sign, but at the same time they may lack the means to construct an account in terms of intentionality. Nevertheless, they often wish to talk meaningfully about an animal sending a signal or even deceiving another organism. In these difficult situations it would be very useful to have a method by which one could draw the distinction between signals and indices without relying on evolutionary to communicate about state S but in this case is used in a different state. history or speakers' intentions.
In the present paper we develop an alternative methodology for capturing the distinction between signals and signs, a methodology which relies solely on the game-theoretic structure of the interaction. Utilizing the canonical two-person models of communication that are studied in philosophy, biology, and economics, we define an "informational move." We say a move is informational when you would not do it if the other party already knew everything you do. By making this notion precise we will be able to distinguish between signals and indices without relying on teleology in one guise or another. Formalizing the notion of informational moves also allows us to uncover a communicative flip-side to the production of signals. In contrast to typical signals that serve to reveal information, we describe this new class of behaviors as "information concealing". This represents two improvements on existing theories. First, in some cases it may be easier to determine whether or not something counts as a signal-namely, when the evolutionary or intentional provenance of a behavior may be empirically inaccessible. Second, it is simpler. Most teleosemantic theories require an understanding of the strategic situation (the game being played), as does our approach. But we do not require the kind of knowledge of evolutionary history that is part and parcel of teleosemantics.

5
to mate with him. The peahen has no intrinsic interest in the peacock's tail, only an interest in mating with the high quality male.
This story is summarized in the extensive form game pictured in figure 1.
Here the benefit of mating for the male peacock is 1, but growing a large tail has a cost of c for the high quality male or d for the low quality male. For the female, mating with the high quality male pays 1 while mating with the low quality male pays 0.
If the cost for the low quality peacock to grow a big tail (d) is sufficiently large, and if the cost for the high quality peacock to grow a big tail (c) is sufficiently small, then a Nash equilibrium exists in which the high quality male grows a long tail and the low quality male does not and in which the female chooses only to mate with males with long tails. 6 In this situation, biologists would say that the high quality peacock is signaling his quality to the female.
Under traditional definitions of a signal, if one is to justify such claims about the signaling nature of this game, one must know either the intentions behind the production of the long tail or the evolutionary history of this phenotype. We believe that this distinction can be drawn without reference to either.
Notice that had the peahen been aware-directly-of the peacock's quality, the peacock would not have bothered to grow a long tail. He grows a long tail because it communicates his quality, but only because it does this.
In order to make this informal idea somewhat more precise, we introduce 6 A Nash equilibrium of a game is a stable state where each individual player is doing the best she can given how the other player(s) are behaving.    In this "comparator game," the peahen is aware of the peacock's quality.
As a result neither the high nor the low quality peacock will grow a long tail -doing so would expend resources for no purpose. So, although there is a Nash equilibrium in the original game that involves high quality peacocks producing a large tail, there is no such equilibrium in this comparator game where the peahen starts off just as well informed as the peacock.
This observation identifies what we call an informational move: First gloss. A move m by player 1 is purely informational if it is performed when player 1 knows things that are unknown to player 2, but not performed when player 2 has all of the information that was available to player 1 when player 1 chose m.
We will make this notion mathematically precise in section 5, but first we will turn to another example.

Information concealing moves
In their (now classic) discussion of signaling games Cho and Kreps because by doing so they would reveal that they are wimpy and would then be challenged to a fight. Instead, even the wimpy individuals drink beer in order to disguise their fighting abilities.
We can again consider our test for informational moves. Suppose now that the locals can observe directly whether the visitor is surly or wimpy. In this situation (shown in figure 4) the surly visitor continues to drink beerit is his preferred breakfast after all -while the wimpy visitor now consumes quiche. Under our test, proposed in the previous section, the consumption of beer by the wimpy visitor is an informational move while the consumption of beer by the surly visitor is not.
This informational move is somewhat different from the case of the peacocks presented in the preceding section. A peacock's long tail carries something like Grice's non-natural meaning. One might reasonably say that the long tail non-naturally means that the the male is of high quality. In the beer and quiche game, the converse occurs. Here the consumption of beer by the wimpy guy is not to convey non-natural meaning, but rather to conceal natural meaning. Drinking beer remains in some respects a communicative act, but of a very different sort from growing a long tail in the peacock scenario.
We use the label information concealing moves for actions like drinking beer despite preferring quiche.
At this point, it is instructive to compare our notion of "information concealing" moves to other notions of misinformation and deception in games.
Skyrms [16], for example, defines a signal as misinformation if it raises the probability of a state which is not actual. 7 Notice that in the beer and quiche game, the action of ordering beer has no effect on the probabilities of the visitor being wimpy or surly. So rather than being misinformation, ordering beer is no-information. This is why we have opted to use the term information concealing rather than deception or misinformation-we believe these are genuinely distinct concepts. 8 7 Skyrms is not the only theorist who draws a distinction between informative and misinformative signaling. A similar distinction could be made with other definitions of deception/misinformation. 8 Notice that case of outright lying, like telling a friend that his karaoke singing is beautiful, is under our definition a "signal" even if it is a misinformative or deceptive one.
The signals we considered in the previous section have production costs that are related to their meanings. High quality peacocks can produce long tails at lesser expense than low-quality birds; surly fellows actually enjoy a breakfast of beer instead of quiche. A short tail cannot come to "mean" that the male is of high quality; eating quiche cannot come to "mean" that a diner is surly.
In their purest form, conventional signals have no relation between meaning and production cost [7]. The signals referred to in the previous sections would probably not be considered (fully) conventional-the structure of the interaction prohibits the reversal of the signals. The paradigmatic case of conventional meaning arises in a Lewis signaling game [9]. As in a costly signaling game, this kind of game has two players, a sender and a receiver.
The sender observes the state of the world and sends a signal. The receiver responds to the signal by choosing an act. Unlike the costly signaling case, there is complete common interest between the sender and the receiver.
Moreover, sending a signal is cost-free.
If there are two states, two signals and two acts, the extensive form of the Lewis signaling game is as in Figure 5, where A and B are zero. The choices of moves indicated in the figure constitute a signaling system. In this signaling system, the sender chooses signal a in response to state A, and the receiver chooses the appropriate act for that state. Similarly, in state B the sender signals b and the receiver chooses the appropriate response.
What makes this kind of signaling purely conventional is that a and b have neither any intrinsic meaning nor any structural properties that prevent their means from being reversed. As indicated by the payoffs, the sender might as Notice that this idea does not depend on how small the epsilons are that This definition allows us to apply our idea of informational moves to extensive form games even if their payoff structure is not generic. The idea is essentially the same as the one that underlies Reinhard Selten's concept of a trembling hand perfect equilibrium [13], where one requires of a Nash equilibrium to be the limit of the Nash equlibria of a sequence of payoffperturbed games. 10

Core theory
We begin by introducing the basics of extensive form games. (Unfamiliar readers should consult any introductory textbook on game theory for more precise definitions.) A game in extensive form begins with a rooted tree, a special kind of mathematical graph which has a beginning node and has no cycles. A node that is not at the end of the tree is called a decision node and is labeled with 10 We can create one sequence that makes signal A informational and another sequence that can make signal B informational. So, on our definition both come out as informational in the original non-generic game. a player who moves at that node. "Nature" is treated as a player who makes moves with some fixed probability -these represent non-strategic decisions which influence the outcome of the game. Every node that is at the end of the tree is called a terminal node and is assigned a payoff value for each player -these represent the various ways that the game might end. Some nodes may be collected together in information sets. These information sets represent ignorance by a player. If nodes n and n are both in the same information set, we interpret this as indicating that the player is not informed whether she resides at n or n . There are various constraints on information sets that formally prevent one information set from containing nodes of more than one player, that prevent players from forgetting something they learned earlier in the game, and that prevent players from being ignorant of their own actions.
Because of the complexity of games in extensive form, we will restrict ourselves to a certain class of games known as "action-response games." These games are essentially games where nature chooses from a set of options with a fixed probability. One player is (potentially) given some information about the state chosen by nature. This player can then take a move. A second player (potentially) observes some information about the state and (potentially) some information about the first player's action and then takes an action herself. After this the game terminates. Formally, an actionresponse game is any game with two players plus a move by nature such that every potential path through the game tree features first a move by nature, second a move by player 1, and finally a move by player 2-no more and no less. Is this compatible with what we say above? There, we say that player 2 might also get some information about nature. This has to go though player 1, though, right?
A game G is generic if there is a unique payoff at each terminal node (i.e., there are no payoff ties). For the first part of this section, we will restrict ourselves to considering generic games.
We now wish to consider whether a particular move m by player 1 is an informational move in a generic action-response game G. To do so, we must specify how to construct the appropriate comparator game. Given the formal constraints on information sets alluded to above, the information sets create a partition on each players decision nodes. Let We now construct the comparator game C(G, m) so that the second player knows everything he knew in the original game, plus everything that the first player knows when he makes the move m. To do this, we keep the tree, nodes, and payoffs the same as in G. We also keep the information sets of the first player the same. We alter the information sets of the second player, such that they become the join I 2 ∨ D(m), i.e., the coarsest common refinement of I 2 and D(m). That is, we add the knowledge "we are in state s" to player 2's information set in the most minimal way possible (without telling them anything else in any other state).
A consequence of this procedure is that the comparator game is defined relative to a particular move m. When evaluating different moves, different comparator games will be constructed that will have different informational structures. With this in hand, we can now provide a formal definition of an informational move in a generic game: Definition Let G be a generic action response game in extensive form. A move m by player 1 in G is an informational move if m is performed with positive probability in some Nash equilibrium of G but is not performed with positive probability in any subgame perfect Nash equilibrium in C(G, m).
This definition is, again, satisfied by the peacock example. In the original game there is a subgame perfect Nash equilibrium where the peacock grows a long tail, but no subgame perfect Nash equilibrium where the peacock does so in the comparator game. Similarly for the other games we have discussed.
As can be seen from our informal definitions above, this way of understanding informational moves has a certain counterfactual quality to it. One can think of G as representing the situation in the actual world and C(G, m) as representing the closest possible world where player 2 has all the information available to player 1 when player 1 chose m.
With this interpretation in hand, we will now discuss some of the assumptions that underly our definition of informational moves. First, we are presuming that in the actual world we are considering a situation where all players are in a Nash equilibrium when they play this game. Undoubtedly there are many situations in this world where players are out of equilibrium.
Because of the relatively weak assumptions of Nash equilibrium, in order to be out of equilibrium the players must either be irrational or have incorrect assumptions about one another's behavior. In the biological context, to be out of equilibrium means that the population must be subject to change due to natural selection.
We do not think the notion of informational moves in out-of-equilibrium situations are particularly helpful, because it might be the case that player 1 thought she was sending information to player 2, but player 2 thought otherwise. Here we believe there is not a clear fact of the matter regarding the informational status of the situation.
Second, we argue that the game C(G, m) characterizes the strategic situ-ation facing the players in the closest possible worlds where player 2 has all the information available to player 1 when she takes move m. Here we equate "closest possible world" with minimum modification of the game tree. While there might be situations where this isn't true, we believe that it should be uncontroversial for most situations of interest.
Finally, we presume that in the closest possible world C(G, m) players are playing a subgame perfect Nash equilibrium. Subgame perfect equilibria are a refinement of Nash equilibria where players are not making non-credible threats-they are not promising to take moves that, if forced to, would be irrational.
We utilize this more restrictive solution concept because in a number of cases that we consider the informational move might continue to be played but only because player 2 is threatening to harm both herself and the other player if the move is not taken. We do not believe that such irrational commitments are plausible, nor are they helpful in uncovering the phenomena with which we are concerned. That we do not use this more restrictive notion when considering play in the actual world should not be taken as a negative comment against this equilibrium notion, but rather a preference for mathematical generality.
Prior to turning to our distinction between information concealing and revealing moves, we must provide one more definition to handle non-generic cases, like the Lewis signaling game.
Definition Let G be a non-generic action-response game in extensive form. A move m in G is said to be an informational move if there exists a sequence of generic games G n such that G n → G and m is an informational move in every G i .
As in the definition of the comparator game, the sequence of games may be different for different moves m and m .
6 Information revealing or information concealing?
In our discussions of the peacock game and of the beer and quiche game we found that some informational moves, such as the peacock's big tail, can be information revealing while other informational moves, such as the wimp's beer breakfast, can be information concealing. In this section we provide a formal mechanism for determining whether a given move is information revealing, information concealing, or neither.
The basic intuition is as follows. A move is information revealing if it gets you to someplace you wouldn't go in the comparator game; a move is information concealing if it avoids going someplace you would go in the comparator game.
To formalize this, consider an action response game G, a move by player However, in the comparator game C(G, m) the nodes for the peahen that occur at the top of the picture, where the peacock grows a long tail, will never be reached in any subgame perfect Nash equilibrium of C(G, m). As a result E C (m) will not contain those nodes and the information set at the top of the picture will not be a member of f (E C (m)).
In order to be an information revealing move, we require that nodes be reached in the original game that would not be reached in the comparator. This captures the idea that information is being revealed rather than concealed. We now turn to that latter type of informational move, To illustrate this definition, consider the beer and quiche game. In the original game, G, the only information set that is reached is the one where the visitor orders beer for breakfast. But, in the comparator game, both beer and quiche are ordered. As a result, ordering beer is regarded as an informational concealing move in the beer and quiche game.

Conclusion
Biologists, economists, and researchers in a range of other disciplines commonly study signaling games, using the machinery of game theory to model communication. But what makes a signaling game a signaling game? Or in other words, where in a signaling game is the signal? Any biologist would answer this by appealing to whatever story he or she told to motivate the analysis -"the peafowl game is a signaling game because a high quality peacock has an incentive to communicate his quality to a potential mate." But this is not a game-theoretic answer. Game theory is not about the stories that one tells alongside one's model; it is the about the analysis of formally defined mathematical objects called games. These objects are fully described by the extended form (or even the normal form) of the game. This poses a problem: it is only meaningful to talk about signaling games if some games are signaling games and others are not. And if this is the case, we obviously have to be able to determine which is which from the extended form game itself, not from the story that someone spins to go along with it. This paper presents the start of a theory of informational moves which avoids the use of intentions, teleology, or motivational stories. While not yet fully general, our hope is that this approach might evolve into a fully general theory of signaling in social interaction. This would enable a more clear articulation of the distinction between signals and other types of biological and economic phenomena, and may provide the groundwork for an alternative theory of meaning. Much work in this direction remains, but we hope that this provides an important starting point.