In many strategic situations a player knows something that another player would like to know. A player can have private information. When Neville Chamberlain was negotiating with Adolf Hitler in 1938, he would have liked to have known Hitler’s true intentions. A player makes the best decision they can in light of the information they have, but they would always like to know more about those on the other side of the strategic divide.
In some scenarios, a player may have the opportunity to learn something about the private information that another player knows. For example, consider a bargaining scenario that might occur at a car dealership. How badly the buyer wants to buy the car may be unknown to the dealer, but the dealer may be able to infer something from the initial offer that the buyer makes. Why do you think car dealers often ask what you’re willing to pay for the car? They want you to reveal information about your willingness to pay, while concealing information about the price at which they’re willing to sell the car.
Another example is given in by the attacker-defender game from the previous post. An attacker can be ‘strong’ or ‘weak’ and it’s strength may be unknown to the defender. However, the attacker may signal its strength to the defender either visually (what they are wearing, how well they are armed, etc.) or through their reputation (have they overcome other defenders in the past, etc.). An attacker can signal their type to the defender, and the defender can act appropriately.
In this post we model and analyse such a scenario. The basic model is known as a signalling game and involves two players: the sender and the receiver. The sender has a certain type, which is given by Nature. The sender’s type is private information to her and is thus unknown to the receiver. The sender chooses an action and may thereby be “signalling” or “sending” information to the receiver. The receiver observes the sender’s action and then chooses an action himself. However, what action is best for the receiver depends on the sender’s type. (It may also depend on the sender’s action.)
To introduce signalling games, and to see what kind of mischief can occur in a signalling game, we consider a situation that you’re apt to find yourself in at some point in life: making a good first impression when dating.
Example I: Dating people from Tinder
You’ve done it. You finally got that Tinder date you’ve been longing for. To this point your chat on Tinder has been pretty minimal… but now it’s your time to shine. You’re meeting up with your date for a few drinks at a bar in town and you need to put your best foot forward. This is the one.
The only advice you’ve been getting from your friends is, ‘just be yourself.’ But you seem to know better. You should absolutely not be yourself; that’s not how you should play this game… So how should you play it? Let’s find out.
Setting the scene
Assume that we have a guy meeting up with a girl (of course it can be two girls meeting up, or two guys meeting up, or whatever, I’m not ruling anything out here). Let’s also assume that the guy is a bit weird and is already completely in love with the girl, and the girl is pretty indifferent about him… so, naturally, the girl has all the power in this scenario.
The girl accepts to meet up with the guy and, on the basis of a few dates, will decide whether to be in a more long-term relationship with him. By long-term relationship I just mean that the girl puts some investment into the guy: meeting each others parents, keeping a toothbrush and clothes at each others’ places, that kind of stuff.
One of the attributes to be learned during this dating period is how ‘nice’ the guy actually is (although, this may not be best characteristic for the girl to observe… we’ll come back to this later!). How ‘nice’ a guy is is just positively related to how much effort he is willing to put into the date: finding good places to go on dates, turning up on time, not burping while eating & drinking, not having several other girls on the side while in the relationship, etc.
To keep matters simple, imagine there are two types of guy in the world: douchebags and gentlemen / true gents. A douchebag is inclined to put minimal effort into the relationship, while a gentleman has inherent natural tendency to put more effort into the relationship.
Suppose that the girl has been messed around by douchebags in the past (and the guy knows this) and wants to build a relationship with a gent. And, regardless of whether he is a douchebag or gentleman, the guy would like to be in a relationship with the girl. She’s great banter.
So what should the guy do to enhance his chances of being in a relationship if he is, in fact, a douchebag? Although he’s not willing to put in much effort into the relationship, even a douchebag may be willing to put in extra effort during the dating period if it will convince the girl that he is gentlemanly and thus worthy of a relationship.
Suppose a douchebag does act like a true gent. The girl, being at least as clever as the lowly douchebag, should recognise that a guy who puts in a lot of effort is not necessarily a gent, but could in fact be a douchebag masquerading as a gent. In that case, the girl cannot infer the guy’s type during the dating period!
Now, consider a guy who is a true gent. If he’s clever, he’ll realise that even a douchebag type will put in a lot of effort in order to avoid conveying the fact that he is a douchebag. So, what should a true gent do? Put in even more effort to being nice! The gent may have to go overboard to distinguish himself from a douchebag. For the girl to distinguish a douchebag from a gent the gent needs to step up his game even more. Geez, life is hard if you’re a nice guy!
Signalling games tend to involve a lot of subtle strategies. So, what is the optimal strategy that the guy should take? What is the optimal strategy that the girl should take? How do we solve these complex games?
Modelling the Tinder date
Thus far in our Tinder date we have players, we have strategies for each player, and we have different types of player.
- Players. A guy (the sender) and a girl (the receiver).
- Strategies. The guy needs to put in some amount of effort during the dating period, and the girl needs to decide whether to keep or dump the guy.
- Types. There are two types of guy in the world: douchebags and gents.
But what about payoffs and beliefs? Determining payoffs and beliefs are important for solving signalling games, so we’ll need to be more specific about these.
Payoffs. We need to assume that all payoffs have some number value that represent utilities. Suppose the payoff to the guy from being in the relationship is 130 and the payoff from being dumped is 70 (not 0 because there are plenty more fish in the sea, blah, blah, blah).
The guy has three options in terms of the amount of effort he can put in—40 units of effort, 60 units of effort, and 80 units of effort—and the personal cost to putting in effort depends on his type, as expressed in the table below. The guy’s overall payoff is the value of being in the relationship (or not), less the personal cost of putting in effort.
We can see from the table that due to the gentleman’s innate tendency to put in more effort that this type of guy incurs less personal cost of putting in more effort. Makes sense.
As noted, the girl would rather form a relationship with a genuinely nice guy as opposed to a douchebag. Accidentally forming a relationship with a douchebag will give her a payoff of 25. Forming a relationship with a gent will provide her a payoff of 100. Continuing to be single provides her a certain payoff of 60.
But what’s the probability that the guy she is dating is a genuinely good guy because, let’s face it, there are a lot of douchebags in the world? From past experience the girl assigns a prior probability of 0.75 (75%) that the guy she is going on a date with is genuinely a douchebag and a prior probability of 0.25 (25%) that the guy is a gent.
These beliefs have the implication that, unless the girl is able to acquire more information about the guy’s type, she’ll not form a relationship with him. Indeed, based on these probabilities the expected payoff from forming a relationship is 0.75 * 25 + 0.25 * 100 = 43.75, which is less than the payoff of 60 from just dumping the guy.
Finally, note that a strategy for the guy assigns an action—40, 60, or 80 units of effort—to each possible type—douchebag or gent—while a strategy for the girl assigns an action—dump or keep—to each possible action of the guy. This is seen in the Bayesian game is depicted below.
This is the structure of the game. Note that the different information sets of the girl are given by dotted lines of different colours… but how do we solve this game?
Solving signalling games
The setup is the easy part, solving it is more difficult. Depending on the complexity of the game, it can be really difficult. Yo, the dating game is a hard game to play! In fact, signalling games in general are pretty intense. So, sorry if this gets a bit complicated.
In the previous post we suggested that to solve a Bayesian game we needed to introduce a solution concept called Bayes-Nash equilibrium. A Bayes–Nash equilibrium is a strategy profile that prescribes optimal behaviour for each and every type of a player, given the other players’ strategies and given beliefs about other players’ types.
Solving signalling games is much like solving a Bayesian game. To do so we need to use a solution concept called a perfect Bayes-Nash equilibrium.
Solution concept: Perfect Bayes-Nash equilibrium
As illustrated by the Tinder dating game, there are three stages to a signalling game.
- First, Nature chooses the sender’s type.
- Second, the sender learns his type and chooses an action.
- Third, the receiver observes the sender’s action, modifies her beliefs about the sender’s type in light of this new information, and chooses an action.
A strategy for the sender assigns an action to each possible type, and a receiver’s strategy assigns an action to each possible action of the sender. The proposed method for solving such a game goes under the grandiose title of perfect Bayes–Nash equilibrium.
Perfect Bayes–Nash equilibrium is founded on two key concepts: sequential rationality and consistent beliefs. Sequential rationality means that, at each point in a game, a player’s strategy prescribes an optimal action, given her beliefs about what other players will do. In the particular context of a signalling game, sequential rationality requires that a sender’s strategy be optimal for each of her types (just as with Bayes–Nash equilibrium) and that a receiver’s strategy be optimal in response to each of the sender’s possible actions.
Sequential rationality requires optimal behaviour, given beliefs. As you can imagine, beliefs can’t be just any old thing, but rather should be cleverly derived in light of the strategic behaviour of other players. In a signalling game, a receiver starts with a set of beliefs about a sender’s type—which are referred to as his prior beliefs and are the probabilities given by Nature—and then he gets to observe the sender’s action before having to act himself.
Because the sender’s action may contain information about the sender’s type, the receiver then modifies his original beliefs to derive a set of posterior beliefs (or beliefs conditional on the sender’s action).
A receiver has consistent beliefs if his posterior beliefs are consistent with the sender’s acting in her own best interests. In other words, a receiver should ask, “Having observed the sender’s behaviour, what types of sender would act in such a way?”
Given this, let’s think back to our Tinder dating game…
Solving the Tinder dating game
For the perfect Bayes-Nash equilibrium we need to consider three aspects. First we need to consider the optimal strategy to each possible type of guy (sender). Second, we need to consider the girl’s (the receiver’s) optimal strategy to each possible action of the guy. And finally, the consistent beliefs of the girl (the receiver).
With regards to the perfect Bayes–Nash equilibrium of the dating game consider the following collection of strategies and beliefs:
- Guy’s strategy: If douchebag, then put in 40 units of effort. If true gent, then put in 80 units of effort.
- Girl’s strategy: If the guy puts in 40 or 60 units of effort, then do not form a relationship. If the guy puts in 80 units of effort then form relationship.
- Girl’s beliefs: If the guy puts in 40 units of effort, then assign a probability of 1 to him being a douchebag. If the guy puts in 60 units of effort, then assign a probability of 0.6 to him being a gent. If the guy puts in 80 units of effort, then assign a probability of 1 to him being a gent.
To determine whether this is a perfect Bayes–Nash equilibrium, start with the guy’s strategy. Given the girl’s strategy noted above, the payoffs from various actions are shown in the table below.
For example, if the guy is a douchebag and puts in 60 units of effort, his payoff is -5 since he incurs a personal cost of 75 and is dumped (as dictated by the girl’s strategy). Given that the guy is a douchebag, it is indeed optimal for him to put in 40 units of effort, as a payoff of 20 exceeds that of -5 and that of 10. However, if the guy is get, he should put in 80 units of effort. Doing so means a payoff of 50—since he gets a girlfriend—and putting in less effort results in a lower payoff because he will be dumped.
Thus, the guy’s strategy is optimal for both of his types, given the girl’s strategy. Moreover, the girl’s strategy is optimal. Further, the girl’s strategy is then optimal for each possible action of the guy. Accordingly, this scenario is a perfect Bayes–Nash equilibrium!
There are multiple forms of perfect Bayes-Nash equilibrium. We can have a separating equilibrium, a pooling equilibrium, or a semi-separating equilibrium. The dating scenario is an example of a separating strategy—a strategy that assigns a distinct action to each type of player. Hence, the receiver can “separate out” each player’s type from her observed play.
This separating equilibrium provides a nice concluding point: To effectively signal her type, a sender who is of an attractive type may need to distort her behaviour in order to prevent being mimicked by a less desirable type.
However… not all signalling games have a separating equilibrium! We’ll talk more about this with regards the attacker-defender application in the next post!
What life lessons can we take away from our dating game?
I’d like think that there are four main lessons to take away from our Tinder dating game.
- First, signalling is an important aspect in everyday life. Dating is just one example.
- Second, good guys have an incentive to be overly nice and put in an extraordinary amount of effort during the dating phase, which can (as an unintended consequence) put a lot of girls off. Of course, we didn’t model this, but because of this douchebags may win regardless.
- Third, evaluating a person based on their effort level may not be the most efficient strategy. Other characteristics may be better, like… I don’t know… how funny they are. A person is either funny or not; it’s a difficult variable to influence in the dating period.
- Finally, douchebags will always be douchebags. They will only be in relationships with other douchebags, and the circle of life continues. This is especially true with regards the separating equilibrium of the signalling game.
Conclusions & next post
This was a pretty long post and we began our investigation of a concept that we will use in the next post… so, let’s recap!
By now we know two things about games and life in general. One, people have private information. Two, people move sequentially. The significance of sequential moves is that a person who moves first may reveal information about what it is she knows that other players do not. A player who moves second may then be able to glean information about a player’s type from her observed behaviour and use that information in deciding what to do. But if that is what is going on, then the player who moves first may adjust her behaviour with the intent to mislead those who are drawing inferences from it!
An objective of here was to sort out these various forces and identify solutions in which all players are acting in their best interests and no players are fooled. This analysis was conducted in the context of a signalling game, which is the simplest structure that embodies these various features. A signalling game involves a player, known as the sender, who has some private information and moves first by choosing an action. The action is observed by a second player, known as the receiver, who, after updating his beliefs as to the first player’s private information, selects an action as well. The solution concept utilised was a perfect Bayes–Nash equilibrium.
These concepts can be applied to the dating world, in which we saw that a separating equilibrium comes into play and helps the receiver (the girl in our case) identify the type of sender they are faced with.
Next post. We’ll look at signalling games in a deeper way and see how these signalling games can be applied to attacker-defender games. Specifically, we will look at how pooling strategies may come into play and why statistics may be important here.