The normal form of the Entry Deterrence game has two pure Nash equilibria, namely (N, f) and(E, a). However, the standard reasoning goes, the first equilibrium is somewhat unreasonable: after all, if Player 1 were to Enter, Player 2 would not want to fight, ex-post Thus, the(n, f)equilibrium is supported by a non-credible threat. For this reason, we wish to rule it out Reinhard Selten proposed the following. First, note that any non-terminal history h in game with observable actions defines a subgame--the game defined by all histories of which h is a subhistory. For instance, in the simple game of Figure 1, there is a subgame starting at(E), corresponding the histories(E),(E, a) and(E, f). The only other subgame is the whole game, which may be viewed as beginning after 0, the initial history A subgame-perfect equilibrium of the original game is then a profile of strategies which induces Nash equilibria in every subgame. Thus, the (N, f) equilibrium is not subgame- perfect, because f is not a Nash equilibrium(trivially, an optimal strategy) in the subgame tarting at(E) We are ready for a few formal definitions Definition 1 Fix an extensive game with observable actions T=(N, A, H, Z, P, UiieN and a non-terminal history hH\Z. The subgame starting at h is the extensive game with observable actions r(h)=(N, A Zh, Plh, (Uiln)ieN), where Hn=h:(h, h)E H is the set of continuation histories, i.e. seque profiles h' such that(h, h) is a history in H; Zn is defined analogously Phh is the restriction of the player correspondence P to continuation histories: formally, for every h'E Hh, Pln(h)=P((h, h)); and each UiIn is defined analogously Similarly, for any strategy profiles=(si)ieN E S in the game r, shh=(siIn)ieN is the strategy profile induced by s in r(h) by letting sin(h)=si((h, h)) for every h'E H]h and i∈Ph(h Note that, according to the definition, the whole game r corresponds to the subgame r(0) Definition 2 Consider a game r with observable actions. A strategy profile s E s is subgame-perfect equilibrium of r iff, for every non-terminal history hE H\Z, sIh is a Nash equilibrium of r(h) Since r=r(0), any subgame-perfect equilibrium(or SPE for short)is necessarily also a Nash equilibrium. The example given above shows that the converse is false Whenever one introduces a solution concept, the first order of business is to establish the existence of a solution according to the concept under consideration. In finite games with perfect information, this is easy to achieve using the backward induction algorithm. Proposition 0.1 Every finite extensive game with perfect information has a SPEThe normal form of the Entry Deterrence game has two pure Nash equilibria, namely (N, f) and (E, a). However, the standard reasoning goes, the first equilibrium is somewhat unreasonable: after all, if Player 1 were to Enter, Player 2 would not want to fight, ex-post. Thus, the (N, f) equilibrium is supported by a non-credible threat. For this reason, we wish to rule it out. Reinhard Selten proposed the following. First, note that any non-terminal history h in a game with observable actions defines a subgame—the game defined by all histories of which h is a subhistory. For instance, in the simple game of Figure 1, there is a subgame starting at (E), corresponding the histories (E), (E, a) and (E, f). The only other subgame is the whole game, which may be viewed as beginning after ∅, the initial history. A subgame-perfect equilibrium of the original game is then a profile of strategies which induces Nash equilibria in every subgame. Thus, the (N, f) equilibrium is not subgame￾perfect, because f is not a Nash equilibrium (trivially, an optimal strategy) in the subgame starting at (E). We are ready for a few formal definitions. Definition 1 Fix an extensive game with observable actions Γ = (N, A, H, Z, P,(Ui)i∈N ) and a non-terminal history h ∈ H \ Z. The subgame starting at h is the extensive game with observable actions Γ(h) = (N, A, H|h, Z|h, P|h,(Ui |h)i∈N ), where: H|h = {h 0 : (h, h0 ) ∈ H} is the set of continuation histories, i.e. sequences of action profiles h 0 such that (h, h0 ) is a history in H; Z|h is defined analogously; P|h is the restriction of the player correspondence P to continuation histories: formally, for every h 0 ∈ H|h, P|h(h 0 ) = P((h, h0 )); and each Ui |h is defined analogously. Similarly, for any strategy profile s = (si)i∈N ∈ S in the game Γ, s|h = (si |h)i∈N is the strategy profile induced by s in Γ(h) by letting si |h(h 0 ) = si((h, h0 )) for every h 0 ∈ H|h and i ∈ P|h(h 0 ). Note that, according to the definition, the whole game Γ corresponds to the subgame Γ(∅). Definition 2 Consider a game Γ with observable actions. A strategy profile s ∈ S is a subgame-perfect equilibrium of Γ iff, for every non-terminal history h ∈ H \Z, s|h is a Nash equilibrium of Γ(h). Since Γ = Γ(∅), any subgame-perfect equilibrium (or SPE for short) is necessarily also a Nash equilibrium. The example given above shows that the converse is false. Whenever one introduces a solution concept, the first order of business is to establish the existence of a solution according to the concept under consideration. In finite games with perfect information, this is easy to achieve using the backward induction algorithm. Proposition 0.1 Every finite extensive game with perfect information has a SPE. 2