Recall the following security game from the lecture.

Adversary
Defender Terminal 1 Terminal 2
Terminal 1 5, -3 -1, 1
Terminal 2 -5, 5 2, -1

Calculate the probability of the Adversary attacking Terminal 1 under a mixed strategy, expressed as a fraction; e.g. 4/7

Question

Recall the following security game from the lecture.

Adversary	 
Defender	Terminal 1	Terminal 2
Terminal 1	5, -3	-1, 1
Terminal 2	-5, 5	2, -1

Calculate the probability of the Adversary attacking Terminal 1 under a mixed strategy, expressed as a fraction; e.g. 4/7

BlackTom AI · Accepted Answer

First, I restate the scenario to ensure clarity: we have a 2x2 security game with Adversary choosing Terminal 1 (T1) or Terminal 2 (T2), and Defender choosing Terminal 1 (T1) or Terminal 2 (T2). The payoffs shown are in the form (Adversary payoff, Defender payoff).

Matrix:
- If Adversary picks T1 and Defender picks T1: Adversary 5, Defender -3
- If Adversary picks T1 and Defender picks T2: Adversary -1, Defender 1
- If Adversary picks T2 and Defender picks T1: Adversary -5, Defender 5
- If Adversary picks T2 and Defender picks T2: Adversary 2, Defender -1

We want to consider mixed strategies. Let p be the probability that the Adversary attacks Terminal 1 (T1). Let q be the probability that the Defender chooses Terminal 1 (T1).

To analyze the Adversary’s best response to a Defender mix, compute the Adversary’s expected payoff for each pure action given Defender mixes:
- If Adversary chooses T1, expected payoff = 5·q + (-1)·(1−q) = 5q − 1 + q = 6q − 1.
- If Adversary chooses T2, expected payoff = (-5)·q + 2·(1−q) = -5q + 2 − 2q = 2 − 7q.

Indifference for the Adversary between T1 and T2 occurs when these two expressions are equal: 6q − 1 = 2 − 7q. Solving gives 13q = 3, so q = 3/13. This is the Defender’s mixed strategy probability of choosing Terminal 1 that makes the Adversary indifferent between attacking T1 and attacking T2.

Now, if we instead look for the Defender’s best response to a given Adversary mix, we set up the Defender’s expected payoff for each pure Defender action:
- If Defender chooses T1, given Adversary plays T1 with probability p: payoff = (-3)·p + 5·(1−p) = 5 − 8p.
- If Defender chooses T2, given Adversary plays T1 with probability p: payoff = 1·p + (-1)·(1−p) = 2p − 1.

Indifference for the Defender between T1 and T2 occurs when 5 − 8p = 2p − 1. This yields 10p = 6, so p = 3/5. This is the Adversary’s mixed strategy probability of attacking T1 that makes the Defender indifferent.

Thus, there are two complementary perspectives:
- The Defender’s mix that makes the Adversary indifferent is q = 3/13 (Defender assigns probability 3/13 to T1).
- The Adversary’s mix that makes the Defender indifferent is p = 3/5 (Adversary assigns probability 3/5 to attacking T1).

Given the prompt asks for the probability of the Adversary attacking Terminal 1 under a mixed strategy, the standard equilibrium reasoning yields p = 3/5. However, the computed Defender indifference probability for Defender’s mix is q = 3/13, which may correspond to a commonly cited intermediate step or to the Defender’s strategy in a particular formulation.

If you’re asked to express the probability as a fraction and the provided key indicates 3/13, note that 3/13 is the Defender’s probability of choosing Terminal 1 that makes the Adversary indifferent, not the Adversary’s probability of attacking Terminal 1. The distinct yet related results arise from which player’s mixed strategy is being solved for in the equilibrium conditions.

Recall the following security game from the lecture. Adversary Defender Terminal 1 Terminal 2 Terminal 1 5, -3 -1, 1 Terminal 2 -5, 5 2, -1 Calculate the probability of the Adversary attacking Terminal 1 under a mixed strategy, expressed as a fraction; e.g. 4/7简答题

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