An incumbent firm publicly announces: 'If any competitor enters our market, we will immediately start a price war that is costly for everyone, including us.' This threat deters entry and constitutes a Nash equilibrium. Why does subgame perfect equilibrium reject this outcome?
AThe threat was announced publicly, making it legally binding and therefore not a game-theoretic construct
BNash equilibrium cannot apply to sequential games with more than two players
CIf entry actually occurred, the incumbent's optimal action would be accommodation — making the threat incredible and irrational to carry out
DThe entrant should simply call the bluff regardless of the threat, so no Nash equilibrium exists
SPE requires rational play at every decision node, including nodes never reached in equilibrium. Even though the price war threat deters entry (so the node where the incumbent must choose is never reached), SPE asks: if that node WERE reached, would the incumbent actually fight? Since fighting is mutually costly and accommodation is better for the incumbent once entry has occurred, the threat is incredible — the incumbent wouldn't carry it out. Nash equilibrium allows incredible off-path threats because those nodes are never tested; SPE eliminates them by demanding rationality everywhere in the game tree.
Question 2 Multiple Choice
In a two-stage sequential bargaining game, backward induction is used to find the subgame perfect equilibrium. What does backward induction specifically require?
ASolving from the first decision node forward, determining each player's best response in sequence
BIdentifying all Nash equilibria first, then eliminating those involving irrational off-path play
CStarting at the final decision nodes and working backward so each earlier choice correctly anticipates what happens downstream
DAssuming both players move simultaneously at each stage and solving the resulting normal-form game
Backward induction starts at the LAST decision nodes of the game tree, determines what players would rationally do there, then folds those choices back into the analysis of earlier nodes. This ensures that each player's strategy at every stage is a best response to what would actually happen later — the defining property of SPE. Forward reasoning (option A) cannot guarantee this because you don't yet know downstream outcomes when analyzing early nodes. The technique works by eliminating the last strategic uncertainty first, then propagating backward.
Question 3 True / False
A Nash equilibrium in a sequential extensive-form game guarantees that most player's strategy is rational at nearly every decision node, including nodes that are not reached during play.
TTrue
FFalse
Answer: False
False — this is precisely the gap that SPE is designed to fill. Nash equilibrium only requires that the overall strategy profile is a mutual best response given the entire game. It says nothing about rationality at off-path nodes (nodes that are never reached in equilibrium play). An incredible threat at an off-path node can sustain a Nash equilibrium because that node is never tested. SPE adds the requirement that behavior within every subgame — including off-path ones — must also constitute a Nash equilibrium. This is why SPE is called a refinement of Nash equilibrium.
Question 4 True / False
Backward induction can typically find a subgame perfect equilibrium for any extensive-form game, regardless of what information players have about prior moves.
TTrue
FFalse
Answer: False
False. Backward induction works cleanly for games of perfect information — where every player observes all prior moves and every decision node is in a singleton information set, making every subtree a well-defined subgame. When players have imperfect information (they don't know exactly where they are in the game tree), many off-path nodes cannot start a properly defined subgame, and backward induction breaks down. SPE alone becomes insufficient; stronger refinements like Perfect Bayesian Equilibrium are needed to handle beliefs at information sets and off-path rationality in imperfect-information games.
Question 5 Short Answer
What is an 'incredible threat,' and why does subgame perfect equilibrium eliminate it while Nash equilibrium does not?
Think about your answer, then reveal below.
Model answer: An incredible threat is a promised action that a player would not rationally execute if the relevant situation actually arose — typically because carrying out the threat would harm the player making it more than the alternative. Nash equilibrium can sustain incredible threats because Nash only requires that strategies are mutual best responses given the full game; if the threat succeeds in deterring the relevant action, the threat node is never reached and its irrationality is never exposed. SPE eliminates incredible threats by requiring Nash equilibrium play within every subgame, including those at off-path nodes — the threatening player's strategy must be optimal at the threat node itself, not just credible as a deterrent.
The entry deterrence game is the canonical example: an incumbent threatens a costly price war to deter entry. This Nash equilibrium logic holds as long as the threat is believed: given the threat, no one enters, so the threat is never called upon. But SPE asks: what would the incumbent do if entry occurred? The optimal response is accommodation, not war. Since the incumbent would not actually fight, the threat cannot rationally deter a forward-looking entrant. SPE forces the analysis to reflect what players would actually do at every stage — not just what they might threaten. This makes SPE essential for analyzing bargaining, market entry, and any sequential interaction where the credibility of commitments is the central strategic question.