Question

. Consider the following decision problem: states -> S S2 S S acts 9 10 12 and the following alternative rankings of the outcomes est best 2-438 2-63 12 -i,-10,-ii 19-3 12 worst 4,2 worst 2,,-1 -12 (that is, for all i,-1,2, (a) Suppose that the agent's ranking is R. For each pair of actions state whether one action dominates T: z. z. ร์เ , 12, with i <j, 2, Zj ). the other. If your claim is that there is no dominance, then explain why and if your claim is that there is dominance then state whether it is strict or weak dominance (with the understanding that if you say "weak" then you mean "weak and not strict") (b) Suppose that the agent's ranking is S. For each pair of actions state whether one action dominates the other. If your claim is that there is no dominance, then explain why and if your claim is that there is dominance then state whether it is strict or weak dominance (with the understanding that if you say "weak" then you mean "weak and not strict") (c) Suppose that the agent's ranking is T. For each pair of actions state whether one action dominates the other. If your claim is that there is no dominance, then explain why and if your claim is that there is dominance then state whether it is strict or weak dominance (with the understanding that if you say "weak" then you mean "weak and not strict").

Answer 1

(a) The agents ranking in R can also be written as :

$\fn_cm Z_{7} \succ (Z_{2} \sim Z_{6}\sim Z_{12}) \succ (Z_{5} \sim Z_{8}) \succ (Z_{1} \sim Z_{10} \sim Z_{11}) \succ Z_{3} \succ (Z_{4} \sim Z_{9})$

Where, $\fn_cm A \sim B$ shows indifference between two alternatives and $\fn_cm A \succ B$ shows that A is preferred to B.

Thus for state-1 we have ,In the state-1 b is a dominant strategy, because it is strictly preferred than all the other strategies.

# Dominant Strategy : A strategy is called dominant when it is better to play this strategy than to any other strategy.

now for state-2,

# Correction : A strategy not an strategy

For state 3,

For state 4,

Note that with all the states combined (state-1,2,3,4) there is no strategy which is always best to play hence for the game as a whole there is no dominant strategy.

We are done with the part (a) of the question. Part (b) is very similar to part (a) this I would encourage you to solve part (b) by yourself for better understanding.

(c)

Our ranking of alternatives in T are as follows,

$\fn_cm Z_{1} \succ Z_{2} \succ Z_{3} \succ Z_{4} \succ Z_{5}\succ Z_{6}.......\succ Z_{10}\succ Z_{11} \succ Z_{12}$

So repeating the same exercise as in part (a) going one by one with each state,

In state-1, from our ranking of alternatives in T

$\fn_cm Z_{1} \succ Z_{5}\succ Z_{9}$

thus act-a corresponding to highest alternative ( Z1 ) in the state-1 is the dominant act/strategy.

In state -2, from our ranking of alternatives in T

$\fn_cm Z_{2} \succ Z_{6}\succ Z_{10}$

thus act-a corresponding to highest alternative ( Z1 ) in the state-2 is the dominant act/strategy.

In state-3, from our ranking of alternatives in T

$\fn_cm Z_{3} \succ Z_{7}\succ Z_{11}$

thus act-a corresponding to highest alternative ( Z1 ) in the state-3 is the dominant act/strategy.

In state-4, from our ranking of alternatives in T

$\fn_cm Z_{4} \succ Z_{8}\succ Z_{12}$

thus act-a corresponding to highest alternative ( Z1 ) in the state-4 is the dominant act/strategy.

Thus note that in all the stages act-a comes out to be the best chosen strategy/act hence, act-a is a dominant strategy/act if we take all the states combined.

Any feedback is appreciated.