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I am stuck on this question from a practice microeconomic theory exam I am working on.

Consider the payoff matrix below, where $x>0$. For what values of x do both players have a dominant strategy? Solve the Nash equilibrium or equilibria in these cases.

Payoff Matrix

Any help with a strategy for solving this would be appreciated.

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may i ask, the reason for X to be >= 50, instead of just x > 50 is that it requires only finding dominant strategy which means it could be weakly dominant or strictly dominant right? – Steve Nov 16 '12 at 7:44
up vote 2 down vote accepted

Strictly dominant strategies $(s_i'')$ satisfy the condition: $$u_i(s_1,\dots,s_i',s_{i+1},\dots,s_n)<u_i(s_1,\dots,s_i'',s_{i+1},\dots,s_n)$$,where $u_i$ is the payoff of every strategy for player $i$ that can be formed from a strategy set $S_i$. In this case, for Firm I (representing the rows of the payoff matrix), a dominant strategy would exist if $50 > 20$ and $90+x > 140$ simultaneoulsy. For Firm II (representing the columns of the payoff matrix), a dominant strategy would exist if $160 > 140 $ and $50 > 90 - x$. Therefore the mutual solution would be: $$x\geq 50$$

As far as the Nash equilibrium is concerned, the only equilibrium possible is the set $(50,50)$ .

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We want that $90 -x \le 50, 90 + x \ge 140$. So $x \ge 50$.

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