# Questions On Game Theoretic Approach

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Chapter 3 - Application: Section 3.1: Game Theoretic Approach This section follows on from the previous chapter as this is a worked through example of a game between a defender and attacker, it is played in a sequential manner where the defender plays first, however all the defenders information is made public unlike the above theory. Example of a one-stage attacker-defender game (Cox, Jr, 2009) Table 1 illustrates a game, which is represented in normal form, with the loss to a defender choosing from two defence options, to either defend target A or B on whether the attacker then attacks either target A or target B. The defender chooses a row for a strategy, and the attacker chooses a column. Both the sets of rows and columns…show more content…
A mixed strategy is where players select probabilistically among their pure strategies. (Cox, Jr, 2009) Here attacker maximises the expected damage to the defender by attacking A with probability: (20 − 110)/((20 − 110) + (10 − 80)) = 90/(90 + 70) = 9/16 = 0.5625 (9) (Cox, Jr, 2009) Here defender minimises the expected loss by defending A with probability: (−20+80)/(−20+80−10+110) = 60/160 = 0.375 (10) (Cox, Jr, 2009) Therefore it can be verified that, for the choices above, the expected return from defending A is 0.5625 × (−10) + (1 − 0.5625) × (−110) = −53.75 and the expected return from defending B is 0.5625 × (−80) + (1 − 0.5625) × (−20) = −53.75 so the mixed-strategy Nash equilibrium condition of equal expected payoffs does satisfy for the defender. Alike calculation can be made for the attacker. Normally, for the general payoff matrix in Table 2 below, the optimal mixed strategy is: Attack A with probability (d1– b1)/[(d1– b1) + (a1 – c1)], else attack B. Defend A with probability (d2 – c2)/[(d2– c2) + (a2 – b2)], else defend B. (Cox, Jr, 2009) Assuming that these two fractions are strictly between 0 and 1, with nonzero denominators and that there is no pure strategy equilibrium. Consequently, for favourable mixed strategies, the attacker expects to cause a loss of −53.75 to the defender. (Cox, Jr, 2009) Source: Table 2 (Cox, Jr, 2009) The defender knows that the attacker’s best response, if the defender select to