Section 4: Dynamic Games of Complete Information
Economics 2010a Annie Liang∗
Recall one limitation of Nash Equilibrium is the possible multiplicity of (unranked) equilibria. Today we discuss an argument for why certain NE may be more “reasonable” than others.
1 A B L R L 2,2 2,2 2 2,2 R 3,1 0,0 A B
3,1 0,0
The strategy profile (L, B) is a NE of this game, but there is a sense in which this equilibrium is not particularly satisfactory. In the occasion in which player 2 is asked to move (an off- equilibrium occurrence), A is a strictly-dominant strategy. Play of B is a non-credible threat, and the refinement of subgame perfection rules out this possibility.
1 Definition
Definition 1.1. A proper subgame G of an extensive-form game T consists of a single node and all its successors in T , with the property that if x0 ∈ G and x00 ∈ h(x0) then x00 ∈ G.
1 1
2
3 4
Figure 1: The game above has four proper subgames.
Definition 1.2. A strategy profile σ of an extensive-form game is a subgame-perfect equi- librium if the restriction of σ to G is a Nash equilibrium of G for every proper subgame G. In words: strategies must define a Nash equilibria in the continuation game beginning at any singleton information set.
Concept Check: Why is every subgame perfect equilibrium also necessarily also a Nash equilibrium?
2 Verification of subgame perfection
Given the potential number of proper subgames, it might seem that verification of subgame perfection would be too unwieldy. The “one-stage deviation” concept allows us to greatly simplify this process. We will examine this principle separately for finite-horizon and infinite- horizon games. Definition 2.1. Strategy profile s satisfies the one-stage deviation condition if there is no player i and no strategy sˆi such that:
t • sˆi agrees with si except at a single h
t • sˆi is a better response to s−i than si conditional on that history h being reached In words: s satisfies the one-stage deviation principle if there is no agent who could gain by deviating from s at one stage and one stage only, reverting subsequently back to s. Theorem 2.2. In a finite multi-stage game with observed actions, strategy profile s is sub- game perfect if and only if it satisfies the one-stage deviation condition.
2 Proof. Clearly the one-stage deviation condition is necessary, or s would not constitute a NE at the continuation game beginning at history ht. Sufficiency is shown as follows: suppose the strategy profile s satisfies the one-stage deviation condition but is not subgame perfect. ˆ tˆ 0 Then, there is some time t and history h such that some player i has a strategy si that is a t 0 better response to s−i than is si in the subgame starting at h . Let t be the largest t such t 0 t t 0 that for some h , si(h ) 6= si(h ). Since the game is finite-horizon, t necessarily exists. Now construct a series of “unravelling” strategies as follows: define r to agree with s0 at all t < t0 and to agree with s from t0 onwards. In every subgame starting at t0, r coincides exactly with s, and differs from s0 in only one stage. Then by assumption that s satisfies the one-stage deviation condition, r must yield at least as high a payoff as s0 at any subgame starting from t0. Since r coincides with s0 prior to t0, r is a profitable deviation from s at time ht. If t0 = t + 1, the contradiction is immediate, since r is different from s at one stage only. If t0 > t+1, construct a new strategy r0 which agrees with s0 at all t < t0 −1 and agrees with s from t0 −1 onwards. Apply the same logic above, and repeat if necessary. The alleged sequence of improvements unravels back to t + 1, and we have the desired contradiction.
We need one additional restriction in order to apply this to infinite games. Let h denote an infinite-horizon history, and let ht denote the initial t periods of this history.
Definition 2.3. A game is continuous at infinity if for each player i the utility function ui satisfies ˜ sup |ui(h) − ui(h)| → 0 as t → ∞. h,h˜ s.t. ht=h˜t This condition requires that events in the distant future are of relative unimportance. A simple example: games with exponential discounting.
Theorem 2.4. In an infinite-horizon multi-stage game with observed actions that is contin- uous at infinity, profile s is subgame perfect if and only if it satisfies the one-stage deviation condition.
Proof. Necessity is again immediate. To prove sufficiency: suppose that strategy s satisfies the one-stage deviation principle and the game is continuous at infinity, but s is not subgame- perfect. Then there exists some stage t and history ht such that a player i could profitably t deviate to a different strategys ˆi in the subgame starting at h . Denote the profitability of this deviation by > 0. Since the game is continuous at infinity, there must exist some strategy 0 0 0 si which agrees withs ˆi up to t > t and agrees with si at all stages from t onwards which improves on si by at least 2 . Such a strategy, however, differs from si at a finite number of stages. By our previous proof, however, no strategy which makes only a finite number of deviations from s can yield a higher utility; hence, we have the desired contradiction.
3