Tit-for-Tat: Sustaining Cooperation

Introduction

In our previous lesson, we analyzed the grim trigger strategy, a severe but effective way to sustain cooperation in an infinitely repeated Prisoner's Dilemma. We saw that if players are sufficiently patient (i.e., their discount factor $δ$ is high enough), the threat of permanent punishment makes cooperation a stable outcome—a Subgame Perfect Nash Equilibrium (SPNE).

However, the grim trigger strategy is absolute. A single misstep, accidental or otherwise, leads to a complete and eternal breakdown of cooperation. This raises a natural question: can cooperation be sustained by a more forgiving strategy?

This lesson focuses on arguably the most famous strategy in game theory: Tit-for-Tat. Our goal is to analyze this strategy to determine when it can sustain cooperation. We will define its simple rules, derive the mathematical conditions under which it forms a Nash Equilibrium, and then, using the concept of subgame perfection, critically examine the credibility of its punishment mechanism.

1. Defining the Tit-for-Tat Strategy

The Tit-for-Tat (TfT) strategy is defined by two simple rules:

Start by cooperating in the first period.
In every subsequent period, do whatever your opponent did in the previous period.

This strategy is:

Nice: It is never the first to defect.
Retaliatory: It immediately punishes a defection with a defection.
Forgiving: It returns to cooperation one period after the opponent does.

The following short video provides a clear definition.

Game Theory 101 (#59): Tit-for-Tat in the Repeated Prisoner's Dilemma

This clip from William Spaniel's 'Game Theory 101' series introduces the Tit-for-Tat strategy and contrasts it with the harshness of Grim Trigger.

Please watch the first 1 minute and 49 seconds of the video. Focus on the simple rules that define Tit-for-Tat.

To make this concrete, consider the following sequence of play where Player 2 is using a Tit-for-Tat strategy.

The Tit-for-Tat Strategy in Action — This table illustrates the Tit-for-Tat mechanism. Player 2's action in each round (starting from round 2) is a direct copy of Player 1's action from the preceding round.

Notice how Player 2 cooperates in Round 2 because Player 1 cooperated in Round 1. Then, Player 2 defects in Round 3 because Player 1 defected in Round 2. Finally, Player 2 cooperates again in Round 4, 'forgiving' Player 1 after they returned to cooperation in Round 3.

2. The Condition for Tit-for-Tat to be a Nash Equilibrium

Now for the core analysis. If both players adopt the Tit-for-Tat strategy, will either have an incentive to deviate? To answer this, we compare the payoff from sticking with the strategy against the payoff from a one-shot deviation.

For consistency, let's use the same payoff matrix from the video, which is a common variant of the Prisoner's Dilemma:

Player 1 \ Player 2	Cooperate	Defect
Cooperate	3, 3	1, 4
Defect	4, 1	2, 2

Here, $R = 3$ , $T = 4$ , $P = 2$ , and $S = 1$ .

Scenario 1: Stick to Tit-for-Tat (Cooperate)

If both players play TfT, they start by cooperating. In period 2, each player sees the other cooperated, so they cooperate again. This continues forever.
The payoff stream is: $3, 3, 3, \dots$
The total discounted payoff is:

V_{Cooperate} = t = 0 \sum \infty 3 δ^{t} = \frac{3}{1 - δ}

Scenario 2: One-Shot Deviation (Defect Today)

Suppose Player 1 deviates in the first period.

Period 1: P1 defects, P2 cooperates (as per TfT rule 1). P1 gets the temptation payoff, $T = 4$ .
Period 2: P2 plays TfT, sees P1 defected, so P2 defects. P1 plays TfT, sees P2 cooperated, so P1 cooperates. The outcome is (C, D). P1 gets the sucker's payoff, $S = 1$ .
Period 3: P2 sees P1 cooperated, so P2 cooperates. P1 sees P2 defected, so P1 defects. The outcome is (D, C). P1 gets $T = 4$ .
Period 4: The pattern repeats. The outcome is (C, D), and P1 gets $S = 1$ .

The payoff stream for the deviating player is an alternating sequence: $4, 1, 4, 1, \dots$
The total discounted payoff is:

V_{Deviate} = 4 + 1 δ + 4 δ^{2} + 1 δ^{3} + \dots

To find a closed-form expression for this, we can group the terms:

V_{Deviate} = (4 + 4 δ^{2} + 4 δ^{4} + \dots) + (δ + δ^{3} + δ^{5} + \dots)

V_{Deviate} = 4 (1 + δ^{2} + δ^{4} + \dots) + δ (1 + δ^{2} + δ^{4} + \dots)

The term in the parentheses is a geometric series with first term 1 and ratio $δ^{2}$ . Its sum is $\frac{1}{1 - δ ^{2}}$ .

V_{Deviate} = \frac{4}{1 - δ ^{2}} + \frac{δ}{1 - δ ^{2}} = \frac{4 + δ}{1 - δ ^{2}}

The Condition for Cooperation

A rational player will not deviate if $V_{Cooperate} \geq V_{Deviate}$ .

\frac{3}{1 - δ} \geq \frac{4 + δ}{1 - δ ^{2}}

Recognizing that $1 - δ^{2} = (1 - δ) (1 + δ)$ , we can simplify:

\frac{3}{1 - δ} \geq \frac{4 + δ}{( 1 - δ ) ( 1 + δ )}

Since $0 < δ < 1$ , we can multiply both sides by $(1 - δ) (1 + δ)$ without changing the inequality:

3 (1 + δ) \geq 4 + δ

3 + 3 δ \geq 4 + δ

2 δ \geq 1 ⟹ δ \geq \frac{1}{2}

Thus, for this game, (TfT, TfT) is a Nash Equilibrium if and only if $δ \geq 1/2$ . If players are sufficiently patient, the alternating punishment is severe enough to deter an initial defection.

The following video provides a step-by-step walkthrough of this exact derivation. For a purely text-based derivation, which you might also appreciate, you can consult the resource [PDF] chapter 14: repeated prisoner's dilemma.

Game Theory 101 (#59): Tit-for-Tat in the Repeated Prisoner's Dilemma

This segment of the video mathematically derives the condition on the discount factor \delta for Tit-for-Tat to be a Nash equilibrium.

Watch from 01:49 to 07:46. Focus on how the payoff streams for cooperation and deviation are constructed and how the geometric series formulas are applied to solve the final inequality.

A quick note: For this specific payoff matrix, the threshold $δ \geq 1/2$ is the same as we found for Grim Trigger. As the video mentions, this is a coincidence, not a general rule. Changing the payoffs would lead to different thresholds for the two strategies.

3. Is Tit-for-Tat's Threat Credible? An SPNE Analysis

We've established that TfT can be a Nash Equilibrium. But is it a Subgame Perfect Nash Equilibrium? This requires the strategy to be optimal in every possible subgame. The crucial test is in the punishment phase: is the threat of retaliation credible?

Let's imagine we are in a subgame that begins just after Player 1 deviated.

History: In the last period, the outcome was (D, C) - Player 1 defected, Player 2 cooperated.
TfT's Prescription: Player 2 must now retaliate by defecting. Player 1, having seen Player 2 cooperate, must now cooperate. The prescribed action profile is (C, D). If both follow TfT from this point on, the payoffs will alternate (1,4), (4,1), (1,4), ...

Is it in Player 2's best interest to carry out this punishment? Let's analyze Player 2's choice.

Scenario A: Player 2 Follows TfT (Retaliates)
Player 2 defects. The outcome is (C, D). Payoff for P2 is 4. The game continues in the alternating punishment cycle.
P2's payoff stream is: $4, 1, 4, 1, \dots$
The discounted value from this subgame is: $V_{Punish} = \frac{4 + δ}{1 - δ ^{2}}$ .

Scenario B: Player 2 Deviates from TfT (Forgives)
Player 2 is supposed to defect, but instead "forgives" Player 1 and cooperates. Player 1 is already playing Cooperate.
The outcome is (C, C). Payoff for P2 is 3.
Because both players cooperated, the game is now back on the main path of cooperation. They will cooperate forever.
P2's payoff stream is: $3, 3, 3, 3, \dots$
The discounted value from this subgame is: $V_{Forgive} = \frac{3}{1 - δ}$ .

Player 2's threat to retaliate is credible only if $V_{Punish} \geq V_{Forgive}$ .

\frac{4 + δ}{1 - δ ^{2}} \geq \frac{3}{1 - δ}

Solving this inequality (by multiplying by $1 - δ^{2}$ ) gives:

4 + δ \geq 3 (1 + δ)

4 + δ \geq 3 + 3 δ

1 \geq 2 δ ⟹ δ \leq \frac{1}{2}

The Contradiction

Here lies the central problem with Tit-for-Tat.

To deter the initial defection, we need $δ \geq 1/2$ .
To make the punishment credible, we need $δ \leq 1/2$ .

For Tit-for-Tat to be a Subgame Perfect Nash Equilibrium, both conditions must hold. This is only true for the "knife-edge" case where $δ = 1/2$ . If $δ > 1/2$ , cooperation is so attractive that a player would rather forgive a defection and return to mutual cooperation than carry out the alternating punishment. The threat is not credible.

The following video is dedicated entirely to this crucial point.

Game Theory 101 (#60): Tit-for-Tat Isn't Subgame Perfect

This follow-up video explains precisely why Tit-for-Tat fails the test of subgame perfection.

Watch from 02:04 to 07:08. This segment clearly lays out the two conflicting conditions on the discount factor, revealing why TfT is generally not an SPNE.

For a rigorous text-based proof of this same result, you can refer to the document [PDF] chapter 14: repeated prisoner's dilemma under the section titled "Tit-for-tat Strategy. (Section 14.10.3)". It systematically checks the incentives in all possible subgames.

Conclusion

In this lesson, we have moved beyond the simple definition of Tit-for-Tat to a rigorous analysis of its strategic properties. While it is an intuitive and often effective strategy, its theoretical foundation is more complex than it first appears.

Key Takeaways:

Tit-for-Tat (TfT) is a forgiving strategy that begins with cooperation and then mimics the opponent's previous move.
In a repeated Prisoner's Dilemma, a pair of TfT strategies can form a Nash Equilibrium if players are sufficiently patient (in our example, $δ \geq 1/2$ ).
The punishment phase of TfT is an alternating sequence of cooperation and defection.
This punishment is not credible if players are too patient ( $δ > 1/2$ ). When the future is highly valued, returning to a state of perpetual cooperation is more appealing than carrying out the punishment.
Because its threat is often not credible, Tit-for-Tat is generally not a Subgame Perfect Nash Equilibrium.

While Grim Trigger is an SPNE, its harshness is a drawback. While TfT is forgiving, its punishment threat can be non-credible. This reveals a subtle tension between the severity of a punishment and its credibility.

In our next lesson, we will broaden our perspective. We have seen how specific strategies can support specific outcomes (like mutual cooperation). But what is the entire set of outcomes that can be sustained in an infinitely repeated game? This will lead us to one of the most powerful results in game theory: the Folk Theorem.

Recommended resources