Cost of moves and gains for being in control ◮ Moves aren’t for free! ◮ Player i pays k i points per move: Defender pays k 0 , Attacker pays k 1 ◮ Being in control yields gain! ◮ Player earns one point for each second he is in control. 55
How well are you playing? (Notation) ◮ Let N i ( t ) denote number moves by player i up to time t . His average rate of play is α i ( t ) = N i ( t ) / t . 56
How well are you playing? (Notation) ◮ Let N i ( t ) denote number moves by player i up to time t . His average rate of play is α i ( t ) = N i ( t ) / t . ◮ Let G i ( t ) denote the number of seconds player i is in control, up to time t . His rate of gain up to time t as γ i ( t ) = G i ( t ) / t . 57
How well are you playing? (Notation) ◮ Score (net benefit) B i ( t ) up to time t is TimeInControl - CostOfMoves: B i ( t ) = G i ( t ) − k i · N i ( t ) ◮ Benefit rate is β i ( t ) = B i ( t ) / t = γ i ( t ) − k i · α i ( t ) ◮ Player wishes to maximize β i = lim t →∞ β i ( t ) . 58
Movie of F L I P I T Game – Global View 59
Movie of F L I P I T Game – Defender View 60
How to play well? 61
Non-Adaptive Play 62
Non-adaptive strategies ◮ A non-adaptive strategy plays on blindly, independent of other player’s moves. 63
Non-adaptive strategies ◮ A non-adaptive strategy plays on blindly, independent of other player’s moves. ◮ In principle, a non-adaptive player can pre-compute his entire (infinite!) list of moves before the game starts. 64
Non-adaptive strategies ◮ A non-adaptive strategy plays on blindly, independent of other player’s moves. ◮ In principle, a non-adaptive player can pre-compute his entire (infinite!) list of moves before the game starts. ◮ Some interesting non-adaptive strategies: 65
Non-adaptive strategies ◮ A non-adaptive strategy plays on blindly, independent of other player’s moves. ◮ In principle, a non-adaptive player can pre-compute his entire (infinite!) list of moves before the game starts. ◮ Some interesting non-adaptive strategies: ◮ Periodic play 66
Non-adaptive strategies ◮ A non-adaptive strategy plays on blindly, independent of other player’s moves. ◮ In principle, a non-adaptive player can pre-compute his entire (infinite!) list of moves before the game starts. ◮ Some interesting non-adaptive strategies: ◮ Periodic play ◮ Exponential (memoryless) play 67
Non-adaptive strategies ◮ A non-adaptive strategy plays on blindly, independent of other player’s moves. ◮ In principle, a non-adaptive player can pre-compute his entire (infinite!) list of moves before the game starts. ◮ Some interesting non-adaptive strategies: ◮ Periodic play ◮ Exponential (memoryless) play ◮ Renewal strategies: iid intermove times 68
Periodic play Player i may play periodically with rate α i and period 1 /α i 69
Periodic play Player i may play periodically with rate α i and period 1 /α i E.g. for α 0 = 1 / 3, we might have: t 70
Periodic play Player i may play periodically with rate α i and period 1 /α i E.g. for α 0 = 1 / 3, we might have: t It is convenient to assume that periodic play involves miniscule amounts of jitter or drift; play is effectively periodic but will drift out of phase with truly periodic. 71
Adaptive play against a periodic opponent An adaptive Attacker can easily learn the period and phase of a periodic Defender, so that periodic play is useless against an adaptive opponent, unless it is very fast. Examples: ◮ a sentry make his regular rounds ◮ 90-day password reset 72
Periodic Attacker Theorem If Attacker moves periodically at rate α 1 (and period 1 /α 1 , with unknown phase), then optimum non-adaptive Defender strategy is ◮ if α 1 > 1 / 2 k 0 , don’t play(!), ◮ if α 1 = 1 / 2 k 0 , play periodically at any rate α 0 , 0 ≤ α 0 ≤ 1 / 2 k 0 , ◮ if α 1 < 1 / 2 k 0 , play periodically at rate � α 1 α 0 = > α 1 2 k 0 73
Graph for Periodic Attacker and Periodic Defender ( k 0 = 1 , k 1 = 1 . 5 ) α 1 2 3 1 2 1 3 1 6 α 0 1 1 1 2 6 3 2 3
Graph for Periodic Attacker and Periodic Defender ( k 0 = 1 , k 1 = 1 . 5 ) α 1 2 3 1 2 1 3 1 6 α 0 1 1 1 2 6 3 2 3
Graph for Periodic Attacker and Periodic Defender ( k 0 = 1 , k 1 = 1 . 5 ) α 1 1 if α 1 > 2 k 0 Attacker too fast for Defender 2 3 1 2 1 3 1 6 α 0 1 1 1 2 6 3 2 3
Graph for Periodic Attacker and Periodic Defender ( k 0 = 1 , k 1 = 1 . 5 ) α 1 2 3 1 2 1 3 1 6 α 0 1 1 1 2 6 3 2 3
Graph for Periodic Attacker and Periodic Defender ( k 0 = 1 , k 1 = 1 . 5 ) α 1 2 1 if α 1 = 3 2 k 0 Defender can play with 0 benefit 1 2 1 3 1 6 α 0 1 1 1 2 6 3 2 3
Graph for Periodic Attacker and Periodic Defender ( k 0 = 1 , k 1 = 1 . 5 ) α 1 2 3 1 2 1 3 1 6 α 0 1 1 1 2 6 3 2 3
Graph for Periodic Attacker and Periodic Defender ( k 0 = 1 , k 1 = 1 . 5 ) α 1 2 3 1 if α 1 < 2 k 0 1 2 Defender maximizes benefit with � α 0 = α 1 1 2 k 0 3 1 6 α 0 1 1 1 2 6 3 2 3
Graph for Periodic Attacker and Periodic Defender ( k 0 = 1 , k 1 = 1 . 5 ) α 1 2 3 1 Optimal Attacker play 2 1 3 1 6 α 0 1 1 1 2 6 3 2 3
Graph for Periodic Attacker and Periodic Defender ( k 0 = 1 , k 1 = 1 . 5 ) α 1 2 3 1 Optimal Attacker play 2 1 3 1 6 α 0 1 1 1 2 6 3 2 3
Graph for Periodic Attacker and Periodic Defender ( k 0 = 1 , k 1 = 1 . 5 ) α 1 2 3 1 Optimal Attacker play 2 1 3 1 6 α 0 1 1 1 2 6 3 2 3
Graph for Periodic Attacker and Periodic Defender ( k 0 = 1 , k 1 = 1 . 5 ) α 1 2 3 1 Optimal Attacker play 2 1 3 1 6 α 0 1 1 1 2 6 3 2 3
Graph for Periodic Attacker and Periodic Defender ( k 0 = 1 , k 1 = 1 . 5 ) α 1 2 3 1 2 1 3 Nash equilibrium at ( α 0 , α 1 ) = ( 1 3 , 2 9 ) 1 6 α 0 1 1 1 2 6 3 2 3
Graph for Periodic Attacker and Periodic Defender ( k 0 = 1 , k 1 = 1 . 5 ) α 1 2 3 1 2 1 3 Nash equilibrium at ( α 0 , α 1 ) = ( 1 3 , 2 9 ) 1 ( γ 0 , γ 1 ) = ( 2 3 , 1 3 ) 6 ( β 0 , β 1 ) = ( 1 3 , 0 ) α 0 1 1 1 2 6 3 2 3 86
Exponential Attacker If Attacker plays exponentially with rate α 1 , then his moves form a memoryless Poisson process; he plays independently in each interval of time of size dt with probability α 1 dt Probability that intermove delay is at most x is 1 − e − α 1 x For α 1 = 0 . 5, we might have: t 87
Graph for Exponential Attacker and Defender) ( k 0 = 1 , k 1 = 1 . 5 ) α 1 1 2 3 1 3 α 0 1 2 1 3 3
Graph for Exponential Attacker and Defender) ( k 0 = 1 , k 1 = 1 . 5 ) α 1 1 2 3 1 3 α 0 1 2 1 3 3
Graph for Exponential Attacker and Defender) ( k 0 = 1 , k 1 = 1 . 5 ) α 1 Attacker too fast if α 1 > 1 1 2 3 1 3 α 0 1 2 1 3 3
Graph for Exponential Attacker and Defender) ( k 0 = 1 , k 1 = 1 . 5 ) α 1 1 2 3 1 3 α 0 1 2 1 3 3
Graph for Exponential Attacker and Defender) ( k 0 = 1 , k 1 = 1 . 5 ) α 1 1 2 3 Optimal Defender play for α 1 < 1 � α 0 = α 1 k 0 − α 1 1 3 α 0 1 2 1 3 3
Graph for Exponential Attacker and Defender) ( k 0 = 1 , k 1 = 1 . 5 ) α 1 1 2 3 Optimal Attacker play 1 3 α 0 1 2 1 3 3
Graph for Exponential Attacker and Defender) ( k 0 = 1 , k 1 = 1 . 5 ) α 1 1 2 3 Optimal Attacker play 1 3 α 0 1 2 1 3 3
Graph for Exponential Attacker and Defender) ( k 0 = 1 , k 1 = 1 . 5 ) α 1 1 2 3 1 3 Nash equilibrium at ( α 0 , α 1 ) = ( 6 25 , 4 25 ) α 0 1 2 1 3 3
Graph for Exponential Attacker and Defender) ( k 0 = 1 , k 1 = 1 . 5 ) α 1 1 2 3 1 3 Nash equilibrium at ( α 0 , α 1 ) = ( 6 25 , 4 25 ) ( γ 0 , γ 1 ) = ( 3 5 , 2 5 ) α 0 ( β 0 , β 1 ) = ( 9 25 , 6 25 ) 1 2 1 3 3 96
Renewal Strategies A renewal strategy is one with iid intermove delays for player i ’s moves: Pr ( delay ≤ x ) = F i ( x ) for some distribution F i . Renewal strategies form a very large class of (non-adaptive) strategies; periodic, exponential, etc. are special cases... Origin of term: player’s moves form a renewal process . 97
Optimal (renewal) play against a renewal strategy. One of our major results is the following: Theorem The optimal renewal strategy against any renewal strategy is either periodic or not playing. 98
Proof notes Average time between buses � = Average waiting time for a bus 99
Proof notes Average time between buses � = Average waiting time for a bus Proof considers size-biased interval sizes... 100
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