What is Sequential Rationality in Dynamic Games?

Imagine you’re playing chess, and your opponent threatens to capture your queen if you move your knight. But wait-would capturing your queen actually be in their best interest at that moment? Maybe it would leave their own king vulnerable. This is the essence of sequential rationality: ensuring that every move in a game makes sense not just at the beginning, but at every single decision point along the way.

In game theory, sequential rationality is a fundamental principle that requires players to choose optimal strategies at every stage of a dynamic game, not merely at the outset. This concept helps us identify which strategic plans are truly credible and which are based on empty threats that players would never actually carry out.

What does it mean to be sequentially rational? 🔗

Sequential rationality demands that a player’s strategy remains optimal at every information set-every point in the game where they must make a decision. A strategy is sequentially rational when it maximizes a player’s expected payoff given their beliefs about where they are in the game and what other players will do next.

Think of it this way: if you’re following a strategy that tells you to do something foolish later in the game (something that hurts you more than helps), then your opponent shouldn’t believe you’ll actually do it. A sequentially rational strategy eliminates such incredible threats by requiring that every action specified makes sense when the time comes to actually take it.

This differs from simpler equilibrium concepts like Nash equilibrium, which only requires that strategies be optimal at the game’s beginning. Sequential rationality imposes the stronger requirement that strategies remain credible throughout the entire game, ensuring that threats and promises are believable because players would genuinely want to follow through on them.

Understanding continuation games 🔗

To check whether a strategy is sequentially rational, game theorists use the concept of a continuation game. A continuation game represents everything that happens from a specific information set forward-all the possible moves, responses, and outcomes that could unfold from that point onward.

Here’s the important distinction: a continuation game is not necessarily the same as a subgame. While subgames must begin at a point where a player has complete information about all prior moves (what’s called a singleton information set), continuation games can start even when there’s uncertainty about previous actions. This makes continuation games a more flexible tool for analyzing sequential rationality in complex situations.

For example, imagine you’re at a decision point but you’re unsure whether your opponent chose action A or action B earlier. You’re in a continuation game, but not in a proper subgame. Still, you need to decide what to do next, and sequential rationality requires that your choice be optimal given your beliefs about which path led you here.

Evaluating optimal actions in continuation games 🔗

To determine if an action is sequentially rational, we calculate the expected payoff for each possible choice at an information set. The player must select the action that maximizes their expected payoff given their beliefs about where they are in the game tree and what will happen next.

This calculation involves weighing different outcomes by their probabilities. If you believe there’s a 70% chance your opponent played Left and a 30% chance they played Right, you compute your expected payoff from each available action using these probabilities, then choose the action with the highest expected value.

How beliefs shape sequential rationality 🔗

Beliefs play a crucial role in sequential rationality, especially in games with incomplete or imperfect information. When you can’t observe all previous moves, you must form beliefs-essentially educated guesses represented as probability distributions-about which node in your information set has actually been reached.

These beliefs aren’t arbitrary. Whenever possible, they should be updated using Bayes’ rule, which is the mathematically correct way to revise probabilities based on new information. When you observe an action, you should update your beliefs about unobserved factors in a way that’s consistent with rational play by your opponent.

For instance, if your opponent has two types-aggressive and cautious-and the aggressive type is much more likely to attack early, then observing an early attack should lead you to believe your opponent is probably aggressive. Sequential rationality requires that your response be optimal given this updated belief.

Eliminating implausible equilibria with sequential rationality 🔗

One of the most powerful applications of sequential rationality is its ability to rule out equilibria that rely on non-credible threats. Consider a simple game where Player 2 faces a choice between Left and Right at some information set. Suppose Left always gives a higher payoff than Right, regardless of what Player 2 believes about how the game reached that point.

If sequential rationality holds, Player 2 must always choose Left-there’s no belief that would make Right optimal. This simple observation can eliminate entire classes of equilibria. For example, in games where one Nash equilibrium relies on a player threatening to take an action that would hurt them more than their opponent, sequential rationality reveals this threat as incredible and eliminates that equilibrium from consideration.

A classic example: the entry deterrence game 🔗

Imagine a game where an entrant must decide whether to enter a market, and an incumbent can respond by either fighting (costly for both) or accommodating (sharing the market). One Nash equilibrium involves the entrant staying out because the incumbent threatens to fight. But would fighting actually be rational for the incumbent once entry has occurred?

If fighting costs the incumbent more than accommodation would, then the threat to fight isn’t sequentially rational. The incumbent would prefer to accommodate if entry actually happens, so the entrant should ignore the threat and enter anyway. Sequential rationality eliminates the equilibrium based on the incredible threat, leaving only the equilibrium where the entrant enters and the incumbent accommodates.

Worked examples bringing the concept to life 🔗

Let’s consider two illustrative examples that demonstrate how to compute sequentially rational strategies by comparing expected payoffs based on beliefs.

The little horsey game 🔗

In this game, Player 1 chooses between Up, Middle, or Down. If Player 1 chooses Down, the game ends. Otherwise, Player 2 must choose Left or Right without knowing whether Player 1 chose Up or Middle. Let’s say Player 2 assigns probability p to being at the node following Up.

Player 2’s expected payoff from choosing Left is p × (payoff if Up) + (1-p) × (payoff if Middle). Similarly for Right. Regardless of what p is-even if Player 2 is completely uncertain-suppose Left always yields a higher expected payoff than Right. Then Left is Player 2’s unique sequentially rational strategy.

This conclusion eliminates any equilibrium where Player 2 plays Right, because Right would never be optimal once Player 2’s information set is reached. Player 1, anticipating Player 2’s rational response, can then determine their own optimal initial move.

The gift-giving game 🔗

Consider a game where Player 1 receives either a desirable gift or an undesirable one, then decides whether to offer it to Player 2. Player 2 observes only that a gift is offered, not which type it is. Player 2 must decide whether to accept or reject the wrapped gift.

Let q represent Player 2’s belief that the gift is desirable. Player 2’s expected payoff from accepting is q × (payoff from good gift) + (1-q) × (payoff from bad gift). The expected payoff from rejecting is typically lower but guaranteed.

By comparing these expected payoffs, we can determine Player 2’s sequentially rational strategy. If accepting yields a higher expected payoff for any belief q > 0.5, then Player 2 should accept whenever they believe it’s more likely than not that the gift is good. This belief-dependent optimal response is precisely what sequential rationality requires.

What do you think? Can you identify a situation from your own experience where someone made a threat they wouldn’t actually have carried out if tested? How might understanding sequential rationality change the way you approach strategic decisions in competitive situations?

References

1. https://en.wikipedia.org/wiki/Perfect_Bayesian_equilibrium
2. https://gametheory101.com/courses/game-theory-101/perfect-bayesian-equilibrium/
3. http://slantchev.ucsd.edu/courses/gt/05-extensive-form.pdf
4. https://en.wikipedia.org/wiki/Sequential_equilibrium