2.1. Lewis Signaling Game
The first game is a standard sender–receiver signaling game. This game type has been extensively studied in many different fields, e.g., philosophy [1
], economics [38
], linguistics [40
], and theoretical biology [42
]. In the following, we will call this game type a Lewis signaling (LS) game, after one of its earliest formulations by Lewis [1
A LS game is a game-theoretic model that outlines information transfer between the sender and the receiver. An LS game is given by a tuple , where T is a set of states, each of which represents the private information of the sender; S is a set that contains signals that the sender transfers to the receiver, and R is a set that contains response actions that the receiver can choose. Furthermore, is a utility function that determines how well a state matches a response action. In all of the games that we consider in the article, there is exactly one optimal action for any state, notified by the same indices. More precisely, the utility function is defined as if , or otherwise 0. In this paper, we consider a variant of the LS game that has three states, three signals and three actions: , , and .
One round of the LS game is played as follows: first, a state is randomly chosen. Then, the sender communicates state t by choosing a signal . Afterwards, the receiver chooses a response . Communication is successful if and only if s matches r, which results in an optimal utility of 1 for both players, or otherwise 0.
The game determines the relationship between states and response actions through its utility function, but it does not determine any relationship between signals and states or signals and actions. Thus, as a consequence of the definition of the model itself, signals are meaningless. However, signals can become meaningful due to the regularities in sender and receiver behavior. Such behavior can be described in terms of strategies. A sender strategy is defined by a function , and a receiver strategy is defined by a function . We describe agents’ communicative behavior by a combination of a sender strategy and a receiver strategy. Therefore, a communicative strategy is defined as a strategy pair of sender strategy and receiver strategy , thus .
The LS game entails
sender strategies and
receiver strategies, resulting in 729 communicative strategies. Only 6 strategies guarantee perfect communication. These 6 strategies enable a one-to-one mapping between states and signals. In the Lewisean diction, these strategies are called perfect signaling systems. Figure 1
shows the six strategy pairs that form perfect signaling systems. These are the only strategy pairs that achieve a perfect expected utility of 1 against themselves, which is equivalent to perfect information transfer.
2.2. Context-Signaling Game
The context-signaling (CS) game is an extended version of the LS game. It is defined by a tuple
. It has the same components as the LS game plus a set C
of contextual cues and a probability function
that maps probabilities of states onto contextual cues, as described below. The idea here is that states can correlate with contextual cues, and receiver strategies can access these cues to construe the very same signal differently given different contexts. This allows the receiver to disambiguate signals that are ambiguously used by the sender [7
]. In other words, some ambiguous signaling systems can guarantee perfect information transfer, provided that a reliable contextual cue delivers the necessary additional information (something that is not possible in LS games, where ambiguous signaling systems can never achieve perfect information transfer). Examples of such perfect ambiguous systems will be given below.
In this paper, we consider a variant of the CS game that has three states, three signals, three actions, and two contextual cues: , , , and . Moreover, we reconsider a CS game where the information states occur with the following probabilities:
In other words, the state only appears with , the state only appears with , and the state appears with or , each with the same probability.
To give an idealized example that is represented by this game, one can imagine using alarm signals in the communication of animals such as monkeys. In this simplified example, a group of monkeys uses three alarm signals to distinguish between different predator types, and for each predator type there is a different optimal response action, such as hiding in a bush or climbing a tree. In our example, there are three different types of predators, represented by the information states , and . Accordingly, is the optimal response actions for an attack by , . The relevant contextual cues are daytime () and nighttime () since predator type is only active at daytime, predator type is only active at nighttime and predator type can potentially attack at any time. By assuming daytime and nighttime to be equally likely, this results in the probabilities , as defined above. Finally, three different signals are at the individuals’ disposal: , and . Note that in this example, a perfect ambiguous system would have (i) the sender using the same signal for the daytime predator and the nighttime predator, and (ii) the receiver arriving at the right response action upon this signal by taking into account whether it is daytime or nighttime.
Formally, one round of the CS game is played as follows: first, a contextual cue is chosen randomly. Then, a state is chosen with probability . Then, the sender communicates the given state by choosing a signal . Afterwards, the receiver chooses a response . Importantly, the receiver knows the current contextual cue c and can use this information for adjusting her behavior. Communication is successful if and only if the state matches the response action, which results in an optimal utility of 1 for both players, else 0.
As in the LS game, a CS game’s sender strategy is defined by a function . However, a receiver strategy is defined by a function since the receiver can also make use of the contextual cue to organize her behavioral pattern. Again, we describe agents’ communicative behavior by a combination of sender and receiver strategy .
The CS game has a much greater strategic space than the LS game (concrete numbers below). Moreover, the CS game has two different types of strategies that guarantee perfect communication, which we will call perfect signaling systems (as defined before) and perfect ambiguous systems. Note that the perfect ambiguous systems of the CS game only use two signals, one of which can be successfully disambiguated by the receiver through contextual cues. Figure 2
a shows an exemplary perfect signaling system and Figure 2
b, an exemplary perfect ambiguous system. In total, the CS game, as defined here, has 54 different perfect signaling systems and 54 different perfect ambiguous systems.
2.3. Context Bottleneck Game
The context bottleneck (CB) game is a CS game with a particular property: it has fewer signals than states. Formally, a CB game and its communicative strategies are defined exactly like for the CS game before, with the only difference in that it has a smaller signal space:
. Therefore, without contextual cues, it would be impossible to achieve perfect information transfer since it is impossible to distinguish between
different states with
different signals. However, the CB game entails ambiguous systems that achieve perfect information transfer. All in all, the CB game has two such perfect ambiguous systems, as shown in Figure 3
Moreover, the CB game has non-perfect ambiguous systems that achieve very high communicative success of
. Two of such systems are shown in Figure 3
c,d. For example, in the system in Figure 3
c, communication only fails when
appears in context
. This case appears with a probability
, and the probability that
is given at all is
since contextual cues are drawn randomly. In the remaining cases, which therefore appear with a probability
, communication is always successful. Setting probabilities off against utilities yields
These non-perfect ambiguous systems are relevant for the following study since they are evolutionarily stable (a concept that we introduce below). Note that the CB game and the CS game both have evolutionarily stable non-perfect ambiguous systems, whereas in the LS game only perfect signaling systems are evolutionarily stable. An overview of the three games and their properties is shown in Table 2