Rushes in Large Timing Games Model Monotone Payoffs in Quantile - PowerPoint PPT Presentation

Rushes in Large Timing Games Model Monotone Payoffs in Quantile Axel Anderson, Lones Smith, and Andreas Park Hump- Georgetown, Wisconsin, and Toronto Shaped Quantile Preferences Comparative Statics Applications Fall, 2016 1 / 41

Wisdom of Old Dead Dudes Natura non facit saltus. -Leibniz, Linnaeus, Darwin, Marshall Examples: Model Monotone Tipping points in neighborhoods with “white flight” Payoffs in Quantile Bank runs Hump- Shaped Land run Quantile Preferences gold rush Comparative Statics ♦ Fundamental payoff “ripens” over time peaks at a Applications “harvest time”, and then “rots” This forces rushes, as in plots 2 / 41

Players and Strategies Continuum of identical risk neutral players i ∈ [ 0 , 1 ] . Players choose stopping times τ on [ 0 , ∞ ) Model Monotone Payoffs in Anonymous summary of actions: Q ( t ) = the cumulative Quantile probability that a player has stopped by time τ ≤ t . Hump- Shaped Quantile With a continuum of players, Q is the cdf over stopping Preferences times in any symmetric equilibrium. Comparative Statics At any time t in its support, a cdf Q is either absolutely Applications continuous or jumps, i.e. Q ( t ) > Q ( t − ) . This corresponds to gradual play , or a rush , where a positive mass stops at a time- t atom. 3 / 41

A Simple Payoff Dichotomy Payoffs depend on the stopping time t and quantile q . Model Common payoff at t is u ( t , Q ( t )) if t is not an atom of Q Monotone Payoffs in Quantile If Q has an atom at time t , say Q ( t ) = p > Q ( t − ) = q , Hump- then each player stopping at t earns: Shaped Quantile Preferences � p u ( t , x ) Comparative p − q dx Statics q Applications A Nash equilibrium is a quantile function Q whose support contains only maximum payoffs. 4 / 41

Tradeoff of Fundamentals and Quantile For fixed q , payoffs u are quasi-concave in t , strictly rising from t = 0 (“ripening”) until a harvest time t ∗ ( q ) , and then strictly falling (‘rotting”). Model Monotone Payoffs in uniquely optimal entry time!!!! Quantile Hump- For all times s , payoffs u are either monotone or Shaped Quantile log-concave in q , with unique peak quantile q ∗ ( s ) . Preferences payoff function is log-submodular, eg. u ( t , q )= π ( t ) v ( q ) Comparative Statics ⇒ harvest time t ∗ ( q ) is a decreasing in q Applications ⇒ peak quantile q ∗ ( s ) is decreasing in time s . Stopping in finite time beats waiting forever: s →∞ u ( s , q ∗ ( s )) < u ( t , q ) ∀ t , q finite lim 5 / 41

Purifying Nash Equilibrium Model Monotone Payoffs in To ensure pure strategies , label players i ∈ [ 0 , 1 ] , and Quantile Hump- assume assume that i enters at time Shaped T ( i ) = inf { t ∈ R + | Q ( t ) ≥ i } ∈ [ 0 , ∞ ) , the “generalized Quantile Preferences inverse distribution function” of Q Comparative Statics Applications 6 / 41

Is Nash Equilibrium Credible? Model Because of payoff indifference, our equilibria are Monotone subgame perfect too, for suitable off-path play Payoffs in Quantile - Assume fraction x ∈ [ 0 , 1 ) of players stop by time τ ≥ 0 . Hump- - induced payoff function for this subgame is: Shaped Quantile Preferences u ( τ, x ) ( t , q ) ≡ u ( t + τ, x + q ( 1 − x )) . Comparative Statics Applications - u ( τ, x ) obeys our assumptions if ( τ, x ) ∈ [ 0 , ∞ ) × [ 0 , 1 ) . 7 / 41

Nash Equilibrium is Strictly Credible (Nerdy) Our equilibria are strictly subgame perfect for a nearby game in which players have perturbed payoffs: As in Harsanyi (1973), payoff noise purifies strategies Model Monotone Index players by types ε with C 1 density on [ − δ, δ ] Payoffs in Quantile stopping in slow play at time t as quantile q yields payoff Hump- u ( t , q , ε ) to type ε . Shaped Quantile ε = 0 has same payoff function as in original model: Preferences u ( t , q , 0 ) = u ( t , q ) , u t ( t , q , 0 ) = u t ( t , q ) , u q ( t , q , 0 ) = u q ( t , q ) . Comparative Statics u ( t , q , ε ) obeys all properties of u ( t , q ) for fixed ε , and is Applications log-supermodular in ( q , ε ) and ( t , ε ) ⇒ players with higher types ε stop strictly later For all Nash equilibria Q , and ∆ > 0 , there exists ¯ δ > 0 s.t. for all δ ≤ ¯ δ , a Nash equilibrium Q δ of the perturbed game exists within (L´ evy-Prohorov) distance ∆ of Q . 8 / 41

Payoffs and Hump-shaped Fundamentals harvest time u ( t , q 0 ) Model Monotone Payoffs in Quantile Hump- Shaped Quantile Preferences t ∗ ( q 0 ) 0 t Comparative Statics Applications 9 / 41

Payoffs and Quantile peak quantile falling rising quantile quantile preferences Model preferences Monotone Payoffs in u ( t 0 , q ) u ( t 0 , q ) Quantile u ( t 0 , q ) Hump- Shaped Quantile Preferences Comparative Statics Applications 0 q ∗ ( t 0 ) 0 q 1 0 1 q q 1 hump-shaped monotone quantile preferences quantile preferences 10 / 41

Tradeoff Between Time and Quantile Since players earn the same Nash payoff ¯ w , indifference prevails during gradual on an interval: Model u ( t , Q ( t )) = ¯ w Monotone Payoffs in So it obeys the gradual play differential equation: Quantile Hump- u q ( t , Q ( t )) Q ′ ( t ) + u t ( t , Q ( t )) = 0 Shaped Quantile Preferences The stopping rate is the marginal rate of substitution, i.e. Comparative Statics Q ′ ( t ) = − u t / u q Applications Since Q ′ ( t ) > 0 , slope signs u q and u t must be mismatched in any gradual play phase (interval): Pre-emption phase: u t > 0 > u q ⇒ time passage is fundamentally beneficial but strategically costly. War of Attrition phase: u t < 0 < u q ⇒ time passage is fundamentally harmful but strategically beneficial. 11 / 41

Pure War of Attrition: u q > 0 If u q > 0 always, gradual play begins at time t ∗ ( 0 ) . So the Nash payoff is u ( t ∗ ( 0 ) , 0 ) , and therefore the war of Model attrition gradual play locus Γ W solves: Monotone Payoffs in u ( t , Γ W ( t )) = u ( t ∗ ( 0 ) , 0 ) Quantile Hump- Shaped Quantile Preferences Comparative Statics Applications 12 / 41

Alarm and Panic running average payoffs : V 0 ( t , q ) ≡ q − 1 � q 0 u ( t , x ) dx Fundamental growth dominates strategic effects if : Model Monotone Payoffs in V 0 ( 0 , q ) ≤ u ( t ∗ ( 1 ) , 1 ) (1) max Quantile q Hump- Shaped When (1) fails, stopping as an early quantile dominates Quantile Preferences waiting until the harvest time, if a player is last. Comparative Statics There are then two mutually exclusive possibilities: Applications - alarm when V 0 ( 0 , 1 ) < u ( t ∗ ( 1 ) , 1 ) < max q V 0 ( 0 , q ) - panic when u ( t ∗ ( 1 ) , 1 ) ≤ V 0 ( 0 , 1 ) . Given alarm, there is a size q 0 < 1 alarm rush at t = 0 obeying V 0 ( 0 , q 0 ) = u ( t ∗ ( 1 ) , 1 ) . 13 / 41

Pure Pre-Emption Game: u q < 0 If u q < 0 always, gradual play ends at time t ∗ ( 1 ) . So the Nash payoff is u ( t ∗ ( 1 ) , 1 ) , and therefore: u ( t , Γ P ( t )) = u ( t ∗ ( 1 ) , 1 ) Model Monotone If u ( 0 , 0 ) > u ( t ∗ ( 1 ) , 1 ) , there is alarm or panic ⇒ a time-0 Payoffs in Quantile rush of size q 0 and then an inaction period along the Hump- black line, until time t 0 where u ( q 0 , t 0 ) = u ( 1 , t ∗ ( 1 )) . Shaped Quantile Preferences Comparative Statics Applications 14 / 41

Equilibrium Characterization [Equilibria] Model Monotone Payoffs in With increasing quantile preferences, a war of attrition 1 Quantile starts at the harvest time in the unique equilibrium. Hump- Shaped With decreasing quantile preferences, a pre-emption 2 Quantile Preferences game ends at the harvest time in the unique equilibrium. Comparative With alarm there is also a time-0 rush of size q 0 obeying Statics V 0 ( 0 , q 0 ) = u ( t ∗ ( 1 ) , 1 ) , followed by an inaction phase, and Applications then a pre-emption game ending at t ∗ ( 1 ) With panic, there is a unit mass rush at time t = 0 . 15 / 41

Rushes Purely gradual play requires that early quantiles stop Model Monotone later and later quantiles stop earlier: u t ≶ 0 as u q ≷ 0 Payoffs in Quantile We cannot have more than one rush, since a rush must Hump- include an interval around the quantile peak Shaped Quantile Preferences There is exactly one rush with an interior peak quantile. Comparative By our logic for rushes, we deduce that equilibrium play Statics can never straddle the harvest time. Applications So all equilibria are early , in [ 0 , t ∗ ] , or late , in [ t ∗ , ∞ ) . 16 / 41

Peak Rush Locus A terminal rush includes quantiles [ q 1 , 1 ] . Model Monotone An initial rush includes quantiles [ 0 , q 0 ] . Payoffs in Quantile The peak rush locus secures indifference between Hump- Shaped payoffs in the rush and in adjacent gradual play: Quantile Preferences u ( t , Π i ( t )) = V i ( t , Π i ( t )) Comparative Statics Applications Since “marginal equals average” at the peak of the average, we have q i ( t ) ∈ arg max q V i ( t , q ) , for i = 0 , 1 17 / 41

Early and Late Rushes Model Monotone Payoffs in Quantile Hump- Shaped Quantile Preferences Comparative Statics Applications 18 / 41

Finding Equilibria using the Peak Rush Loci Model Monotone Payoffs in Quantile Hump- Shaped Quantile Preferences Comparative Statics Applications 19 / 41