Signals Concealing Positive Information Can Coexist in the General Beer-and-Quiche Type Games

Alright, let’s talk about communication. Not just your everyday chat, but the deep-down, strategic kind you see everywhere, from animals trying to figure each other out to, well, people deciding what to order for breakfast in a potentially dodgy bar. It’s all about signals, information, and sometimes, a little bit of bluffing or hiding what you’re really about.

Think about the natural world. Animals signal their strength, their availability, their needs. But how do you know if the signal is honest? This is where signaling games come in. They’re a neat way to model these interactions between a “sender” who knows something private about themselves (like their strength or type) and a “receiver” who has to decide how to act based on the sender’s signal. The big challenge for the receiver is that they can’t see the sender’s true nature directly.

The Classic Problem: Costly Signals vs. Cheap Talk

For a long time, a big idea in this field was the “costly signaling theory.” The basic premise is that honest signals are stable because they’re *expensive* for the sender to produce, especially for those who might be faking it. If you’re weak, pretending to be strong by doing a really tough display might cost you too much energy or risk injury. But if you’re genuinely strong, it’s no biggie. So, only the strong can afford the signal, making it reliable. This works for things like elaborate mating displays or even prey signaling to predators that they’ve been spotted and aren’t worth chasing.

But what about situations where signals aren’t necessarily *costly* in the traditional sense? Or where the sender might actually *want* to hide something positive about themselves? This is where things get really interesting, and it brings us to a rather famous example in game theory: the Beer-Quiche game.

Enter the Beer-Quiche Game

Picture this: You’re a traveler, and you find yourself in a rough bar for breakfast. Locals are known to pick fights with visitors. Now, visitors come in two types: “surly” (tough, prefer beer) and “wimpy” (not tough, prefer quiche). The locals (the receivers) don’t know your type, but they *really* don’t want to fight a surly guy, though they’d happily take on a wimpy one. You (the sender) have to order either beer or quiche. Your order is your signal. Based on your order, the locals decide whether to fight or not.

The twist? Surly guys prefer beer, wimpy guys prefer quiche. But *everyone* prefers avoiding a fight over getting their preferred breakfast followed by a brawl. This creates a tension: do you order what you like, or do you order something else if it might trick the locals into not fighting?

This game, while seemingly simple, captures a core dynamic of asymmetric information and potential mismatching between signals and types. The original Beer-Quiche game was used to show how beliefs play a role in sequential games. But what happens when we look at it from a strategic perspective, focusing on the rules of decision-making for both players?

Mapping the Strategies: A 4×4 World

Instead of thinking step-by-step (Nature picks type, Sender signals, Receiver acts), we can represent this game in a “strategic form.” Imagine all the possible rules the sender could follow and all the possible rules the receiver could follow. For our simplified game with two types, two signals, and two actions, this turns into a 4×4 matrix of strategies and their resulting payoffs.

On the sender side, a strategy isn’t just “order beer.” It’s a rule based on their type:

• Strategy 1 (S1): If surly, order beer; if wimpy, order quiche (This is the “honest” strategy).
• Strategy 2 (S2): If surly, order beer; if wimpy, order beer (Wimpy bluffs).
• Strategy 3 (S3): If surly, order quiche; if wimpy, order quiche (Surly hides positive info).
• Strategy 4 (S4): If surly, order quiche; if wimpy, order beer (Everyone orders their less preferred).

And for the receiver, a strategy is a rule based on the signal observed:

• Strategy 1 (R1): If beer, don’t fight; if quiche, fight.
• Strategy 2 (R2): If beer, fight; if quiche, don’t fight.
• Strategy 3 (R3): If beer, don’t fight; if quiche, don’t fight.
• Strategy 4 (R4): If beer, fight; if quiche, fight.

When we analyze this 4×4 game, we look for “Nash equilibria.” These are stable points where neither the sender nor the receiver can improve their outcome by unilaterally changing their strategy, assuming the other player’s strategy stays the same. It’s where the strategies are mutually best responses.

Different Ways to Play the Game

Our analysis of this general game reveals five possible Nash equilibria, falling into three main types:

1. Separating Equilibrium:

This is where the signals are honest. Different types send different signals, and the receiver acts accordingly. In the Beer-Quiche game, this would be the surly guys ordering beer and wimpy guys ordering quiche, and the locals fighting anyone who orders quiche but leaving the beer drinkers alone. This equilibrium happens when the wimpy guys value getting their preferred quiche *so much* that they’re willing to risk the fight that comes with it. In other words, they value their signal preference more than avoiding the receiver’s undesirable action. It’s the classic “honest” outcome, often supported by costly signaling ideas (though here, the “cost” is the risk of a fight).

2. Pooling Equilibria:

Here, all sender types send the *same* signal. This means the signal carries *no* information about the sender’s type.

• Bluffing Pooling: We found equilibria where *all* visitors order beer, regardless of whether they are surly or wimpy. The wimpy guys are bluffing, pretending to be surly. This works if the locals decide *not* to fight anyone who orders beer (R1). This can be stable if surly visitors are common enough that the locals figure it’s safer overall to just leave beer drinkers alone, even though they know some wimpy guys are getting away with it. Both types prefer avoiding the fight (action) over getting their preferred signal (beer/quiche). This aligns with some previous models of bluffing, like in certain crustaceans where weaker individuals mimic the threat displays of stronger ones.
• A Novel Concealing Pooling: And here’s the really cool, new part we identified. We found an equilibrium where *all* visitors order quiche (S3). Yes, even the surly, tough ones! And the locals, surprisingly, decide *not* to fight anyone who orders quiche (R3 strategy, or a mix including it). Why would this happen? It occurs when the surly visitors value avoiding a fight (the receiver’s action) much more than getting their preferred beer (their signal). They are effectively *hiding* their positive attribute (being tough/surly) by sending the signal associated with the “wimpy” type. And the receivers? They’ve learned that when someone orders quiche in this bar, they’re actually quite likely to be tough, so it’s better not to pick a fight. It’s like the signal’s meaning has been flipped or “buried.” This is similar to the “signal-burying” concept where individuals might deliberately make their positive traits less conspicuous (think modesty or anonymous good deeds).

3. Hybrid Equilibrium:

This is a mix. Senders might sometimes send a signal that doesn’t match their type, and receivers might sometimes act in a way that doesn’t perfectly correspond to the signal. In our Beer-Quiche game, this happens when wimpy visitors sometimes order beer (bluffing), and locals sometimes fight beer-drinkers but *always* fight quiche-drinkers. It’s a less predictable scenario where both sides are using mixed strategies. This can occur when wimpy types value avoiding the fight more than their signal preference, and there are enough wimpy types around to make the locals wary.

What Drives These Outcomes?

The existence of these different equilibria really depends on a couple of key things:

• Sender Preferences: Do senders value getting their preferred signal (beer or quiche) more, or do they value getting the receiver’s preferred action (no fight) more? If avoiding the fight is paramount for both types, you’re more likely to see pooling or hybrid strategies where signals might not be honest.
• Type Frequency: How common are the surly vs. wimpy visitors? If surly visitors are very common, locals might decide it’s just safer to assume anyone ordering beer is tough and not fight, making the bluffing equilibrium (S2, R1) stable. If wimpy visitors are common, the dynamics shift.

Our analysis shows that the classic Beer-Quiche scenario, where both types value avoiding the fight more than their signal preference, falls into the category where pooling equilibria are possible. And crucially, it’s where that novel pooling equilibrium (S3, R3/R4 mix) exists – the one where surly guys order quiche and locals don’t fight.

Beyond the Bar: Why This Matters

So, why spend so much time thinking about breakfast choices in a hypothetical bar? Because this game is a stand-in for countless real-world interactions involving asymmetric information and potential deception or concealment.

The finding that signals *hiding positive information* can be a stable outcome is particularly insightful. It’s not just about low-quality individuals bluffing to appear high-quality. High-quality individuals might strategically choose to appear low-key or “bury” their signals if the payoff from avoiding a negative outcome (like a fight, or unwanted attention) outweighs the benefit of openly advertising their quality.

This expands our understanding of communication beyond simple honest signaling or straightforward bluffing. It suggests that complex strategies of concealment can be stable features of communication systems, especially in situations where there’s a partial conflict of interest between the sender and the receiver (like in the Beer-Quiche game, where the sender wants no fight, but the receiver wants to fight wimpy types but not surly ones).

It shows that communication isn’t always about clear, transparent information transfer. Sometimes, the most stable strategy is to be deliberately ambiguous or even misleading about your positive traits, and receivers might learn to interpret these “opposite” signals.

The Takeaway

What we learned from mapping out the Beer-Quiche game in this strategic form is that the world of signaling is richer and more complex than just costly honesty. Bluffing can persist, hybrid strategies are common, and yes, even hiding your good qualities can be a perfectly stable way to navigate interactions.

The existence of that novel pooling equilibrium where high-quality types conceal their strength by sending a “weak” signal, and receivers learn to interpret it paradoxically, is a fascinating demonstration of how receiver strategies can maintain seemingly counter-intuitive signaling behaviors. It highlights that communication stability isn’t just about the sender’s costs, but also about the dynamic interplay and learned responses of the receivers.

Of course, figuring out how these strategies *evolve* over time is the next big question, requiring a look at evolutionary dynamics. But for now, knowing that these different ways of signaling – including concealing positive stuff – can coexist as stable points in the game gives us a much broader picture of how information flows (or doesn’t) in the complex interactions around us.

Source: Springer