Why Sequential Games Need a Stronger Equilibrium Test
In a sequential game, one player acts before another, so an early decision can depend on what the first player expects later players to do. A prediction about a later action is useful only if that action is still in the later player’s interest when the decision point arrives. Subgame-perfect equilibrium provides a disciplined test: the proposed strategies must be optimal not just for the whole game, but in every continuation that could be reached.
This distinction matters because a strategy specifies what a player would do at every decision point, including points that are not reached when the game is played. An off-path action can affect an earlier player’s choice, even though the later player never actually takes it. The central question is whether that contingent action would really be chosen if the relevant point were reached.
For this tutorial, the backward-induction method is used for finite games with perfect information: players know the actions taken earlier, and each decision node has a single player who chooses an action. Payoffs represent players’ preferences over final outcomes, and players choose actions that maximize their own payoffs. At a node with multiple equally good actions, any maximizing action can be part of a solution. If every choice is strict, the backward-induction solution is unique.
Backward Induction and Credibility
Backward induction begins at the last decisions, where a player can compare the final payoffs directly. Once an optimal action is identified, it determines the continuation payoff at the preceding node. The process continues toward the start. An action that is optimal at its decision node is sequentially rational there; a proposed threat that is not optimal at that node is noncredible.
At each final decision node, identify the action that gives the player who moves there the highest payoff.
Carry the selected action’s full payoff pair back to the preceding decision node.
At each earlier node, the player compares the payoffs from the remaining continuation outcomes and chooses the preferred one.
State the selected action at every decision node, including nodes not reached along the predicted path. Check that the resulting choices are optimal in every subgame.
The logic behind the method is that a player at a later decision node can choose among the actions available there, so an equilibrium continuation must give that player a best response within that subgame. If a proposed action is not best, the player can improve by changing only the action in that continuation. The proposed strategies therefore cannot be a Nash equilibrium of that subgame, and hence cannot form an SPE. Working backward uses this requirement at each node.
A threat is credible when carrying it out would be optimal for the player at the decision point where it would be made. A threat is noncredible when the player would prefer a different action if that point were reached. A first mover who anticipates rational play should not base a choice on a noncredible threat. This test concerns incentives within the game as specified; a binding commitment that changes the available choices would require a different game model.
Worked Example: Finding the Equilibrium in a Two-Stage Game
Player A first chooses X or Y. After either choice, Player B chooses one of two responses. The terminal payoff pairs are listed as (A’s payoff, B’s payoff): after X, response \(x_1\) gives \((3,2)\) and \(x_2\) gives \((1,4)\); after Y, response \(y_1\) gives \((4,3)\) and \(y_2\) gives \((2,1)\). Find the SPE and its outcome.
Method: backward induction. After X, B compares the second payoffs: \(2\) from \(x_1\) and \(4\) from \(x_2\). Thus B chooses \(x_2\), giving the continuation payoff pair \((1,4)\). After Y, B compares \(3\) from \(y_1\) and \(1\) from \(y_2\), so B chooses \(y_1\), giving \((4,3)\).
A now compares the first payoffs from those optimal continuations: choosing X gives A \(1\), whereas choosing Y gives A \(4\). A chooses Y. The SPE strategies are therefore: A chooses Y; B chooses \(x_2\) after X and \(y_1\) after Y. The realized outcome is \((4,3)\).
Check and common mistake. The specified response after X is still part of B’s strategy even though X is not reached. Choosing \(x_1\) there would not be optimal for B, since \(4>2\). A common error is to choose A’s initial action before checking B’s later choices, or to omit the off-path response. The backward-induction comparisons show that both continuation choices and the initial choice are optimal.
A Nash Equilibrium Can Rely on a Noncredible Threat
A Nash equilibrium checks whether any one player can gain by changing their strategy while the other player’s strategy is held fixed. It does not necessarily require a player’s prescribed action to be optimal in a subgame that is not reached. This is why a Nash equilibrium can be supported by an off-path threat that would not be carried out.
Worked Example: Testing an Off-Path Threat
A chooses Red or Blue. After Red, B chooses \(r_1\) or \(r_2\); after Blue, B chooses \(b_1\) or \(b_2\). Payoffs are (A, B): Red-\(r_1\) gives \((3,1)\), Red-\(r_2\) gives \((0,0)\), Blue-\(b_1\) gives \((2,0)\), and Blue-\(b_2\) gives \((5,2)\). Consider the strategy profile in which A chooses Red and B chooses \(r_1\) after Red and \(b_1\) after Blue. Is it a Nash equilibrium? Is it subgame-perfect?
Test for Nash equilibrium. Given B’s stated strategy, A gets \(3\) from Red. If A changes to Blue while B still chooses \(b_1\), A gets \(2\). Since \(3>2\), A does not gain by deviating. On the realized Red path, B gets \(1\) from \(r_1\), compared with \(0\) from \(r_2\). B has no profitable deviation at the reached decision point. If B changes only the off-path response after Blue from \(b_1\) to \(b_2\), the outcome remains Red-\(r_1\), so B’s payoff remains \(1\). Thus the profile is a Nash equilibrium of the whole game.
Test for subgame perfection. In the subgame after Blue, B would get \(0\) from \(b_1\) and \(2\) from \(b_2\). Therefore B would choose \(b_2\), not \(b_1\), if Blue were reached. The proposed Blue-\(b_1\) threat is noncredible, so the profile is not an SPE.
For the SPE, B chooses \(r_1\) after Red and \(b_2\) after Blue. A compares the resulting payoffs \(3\) from Red and \(5\) from Blue, so A chooses Blue. The SPE outcome is \((5,2)\). The key distinction is that A’s Nash deviation test in the proposed profile uses B’s specified response \(b_1\), giving A \(2\), not the payoff from Blue-\(b_2\). The off-path response is nevertheless ruled out by the separate subgame test.
Several Decision Stages and Complete Strategies
When a game has more than two stages, the same reasoning applies. A player’s complete strategy may include choices at several separate decision nodes. The equilibrium outcome records only the path actually taken; the strategy profile must also specify optimal behavior in every continuation that is not taken.
Worked Example: Solving a Game with an Intermediate Decision
A first chooses U or D. After U, B chooses L or R. L ends the game with payoffs \((4,2)\). If B chooses R, A chooses \(u\) or \(d\): \(u\) gives \((1,3)\), while \(d\) gives \((3,1)\). After D, B chooses \(l\) or \(r\), which end the game with payoffs \((2,4)\) and \((0,1)\), respectively. Find the complete SPE strategies and the outcome.
Method: solve each continuation from its final decision. If B chooses R after U, A compares \(1\) from \(u\) with \(3\) from \(d\), and chooses \(d\). Therefore the continuation after R has payoffs \((3,1)\). At B’s decision after U, B compares \(2\) from L with \(1\) from choosing R and then receiving A’s optimal response \(d\). Since \(2>1\), B chooses L.
After D, B compares \(4\) from \(l\) with \(1\) from \(r\), and chooses \(l\). A’s initial comparison is then \(4\) from U (because B chooses L) versus \(2\) from D (because B chooses \(l\)). A chooses U.
The SPE strategies are: A chooses U initially and would choose \(d\) if the later node after R were reached; B chooses L after U and \(l\) after D. The realized outcome is U followed by L, with payoffs \((4,2)\). The later choice \(d\) is part of A’s strategy even though B’s optimal choice L means that node is not reached.
Check and common mistake. At B’s choice between L and R, the relevant payoff from R is B’s payoff after A’s optimal response, namely \(1\), not the payoff of \(3\) that A receives. Comparing the wrong player’s payoff can reverse the choice. Each decision must be evaluated using the payoff of the player who acts at that node.
What to Check in an Exam-Style Analysis
A reliable solution separates the path from the full strategy profile. First solve the last decision nodes, then work backward, and finally state all contingent actions. To assess whether a proposed threat is credible, isolate the subgame that begins where the threat would be carried out and compare the acting player’s payoffs there.
| Question | What to test |
|---|---|
| Is the profile a Nash equilibrium of the whole game? | Given the other players’ strategies, can any player gain by changing their complete strategy? |
| Is an off-path action credible? | At the decision node where it would be taken, does it maximize the acting player’s payoff in that continuation? |
| Is the profile subgame-perfect? | Does the profile prescribe a Nash equilibrium in every subgame, including those not reached on the equilibrium path? |
A frequent trap is to call every Nash equilibrium of a sequential game subgame-perfect. The implication goes only one way: every SPE is a Nash equilibrium of the whole game, but some whole-game Nash equilibria fail the subgame test. Another trap is to treat an off-path action as irrelevant because it is not observed in the predicted outcome. It can matter for a Nash deviation calculation, but it must still be optimal in its own subgame to qualify for an SPE.
The method is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. Its practical value is broader than finding one predicted outcome: it makes explicit which conditional actions support that prediction and whether those actions are consistent with the players’ incentives.
Check Your Understanding
For each problem, use the player’s own payoff at each decision node and state off-path actions when relevant.
- A chooses M or N. After M, B chooses \(m_1\), giving \((3,2)\), or \(m_2\), giving \((1,4)\). After N, B chooses \(n_1\), giving \((4,1)\), or \(n_2\), giving \((2,3)\). Find the SPE strategies and outcome. (Answer: B chooses \(m_2\) after M and \(n_2\) after N; A chooses N; outcome \((2,3)\).)
- A chooses Cedar or Pine. After Cedar, B chooses \(c_1\), with payoff \((2,3)\), or \(c_2\), with payoff \((1,1)\). After Pine, B chooses \(p_1\), with payoff \((3,0)\), or \(p_2\), with payoff \((0,2)\). Find the SPE. (Answer: B chooses \(c_1\) after Cedar and \(p_2\) after Pine; A chooses Cedar; outcome \((2,3)\).)
- A chooses East or West. After East, B chooses \(e_1\), with payoff \((4,2)\), or \(e_2\), with payoff \((1,3)\). After West, B chooses \(w_1\), with payoff \((2,4)\), or \(w_2\), with payoff \((5,1)\). Find the SPE strategies and outcome. (Answer: B chooses \(e_2\) after East and \(w_1\) after West; A chooses West; outcome \((2,4)\).)
- A chooses Silver or Gold. After Silver, B chooses \(s_1\), yielding \((4,2)\), or \(s_2\), yielding \((1,0)\). After Gold, B chooses \(g_1\), yielding \((2,0)\), or \(g_2\), yielding \((5,3)\). A chooses Silver, and B’s strategy is \(s_1\) after Silver and \(g_1\) after Gold. Is this profile Nash? Is it subgame-perfect? (Answer: It is Nash: A gets 4 from Silver versus 2 from deviating to Gold given \(g_1\), and B cannot improve on the reached path. It is not subgame-perfect because after Gold B prefers \(g_2\), since \(3>0\).)
- A chooses P or Q. After P, B chooses \(p\), ending with \((3,2)\), or \(q\), after which A chooses \(a\), giving \((1,4)\), or \(b\), giving \((4,1)\). After Q, B chooses \(r\), ending with \((2,3)\), or \(s\), ending with \((0,0)\). Find the complete SPE strategies and outcome. (Answer: After \(q\), A chooses \(b\); after P, B chooses \(p\); after Q, B chooses \(r\); A chooses P; outcome \((3,2)\).)