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Synergy and Dynaco are the only two firms in a specific high-tech industry. They face the following payoff matrix as they decide upon the size of their research budget:a. Does Synergy have a dominant strategy? Explain.b. Does Dynaco have a dominant strategy? Explain.c. Is there a Nash equilibrium for this scenario? Explain. ($Hint$: Look closely at the definition of Nash equilibrium.)
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Yi Chun Lin
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here, we're looking at game theory between two firms and their decision on whether or not to have a large or a small budget. You can see their payoff matrix here for synergy and Dina Co. I'd like to know whether each firm has a dominant strategy or not. So let's start with synergy now to find whether or not they have a dominant strategy. Let's ask ourselves if Synergy believes that Dina Co is going to choose large. Well, what will synergy choose? Well, if Dina Coz choosing a large budget, Synergy is going to make that choice that gives it the greatest profits. Which means it's going to choose a large budget as well because if it chose small it would gain nothing. But now, supposing that Dina Co is choosing small, if that's what synergy thinks, Dina Code is going to do there instead going to choose small over here. So you can see that they don't have a dominant strategy because they're going to change their decision based upon what Dina Co does. Now let's see if Dina Co has a dominant strategy. Well, if Dina Co believes that Synergy is going to choose large, well then Dina co is going to choose large so that they can earn that 30 million compared to zero. Now, if Dina Co believes synergy is going to choose small, Dinozzo is still going to choose large because that 70 million is greater than the 50 million they would get if they chose small, meaning that Dina Co does have a dominant strategy and the dominant strategy is to choose a large budget. Now, how about a nash equilibrium and a nash equilibrium is going to be that decision which gives neither firm and an incentive to deviate, meaning they would be worse off if they chose to unilaterally deviate from that decision. So that's meaning that the other firm hasn't changed. But if they then choose to switch their decision, they're going to lose. So let's figure out if there's a nash equilibrium here, let's start with large, large right here. So if synergy and Dina Co both have a large budget, does synergy have incentive to deviate? Well, if synergy deviates from the large budget to a small budget, they're going to gain zero, so they're not going to deviate from that. Now, if we look at Dinah Co, Dina Co as a $30 million dollar gain here with a large budget, if they choose to instead deviate to a small budget looks like they're gonna earn gains of zero. So this is telling us that we do have a nash equilibrium. And it's at the point where both firms choose to have a large budget.
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