Question 1
Suppose rock beats scissors in which case the points earned are two and not one, then we would expect a significant change to take place in the entire play. This would require that even for scissors and paper, the points be increased by an equal amount in order to ensure equilibrium in all cases. For the play to be interesting and highly appealing there should be fairness all round so that the winning parties usually get what the others would get if they were also to win. This makes the game more interesting and even reliable in making certain determinations, which have significant implications for the players . Essentially, each of these three items used in the play have significant implications and meanings in the play as such, the significance of each of these items is noted by the number of times that it is played in the time that is used for the game. If the time is altered then there should be certain expectations that the results to be achieved do not actually reflect the actual implications in the game. Playing the paper one-half of the time and rock and scissors dropping to one-fourth has a significant implication.
The paper has a far greater significance and the more its wins in the game the better the situation for which it is being played. Still, it is also difficult to organize the game so that it is easy to win for the paper item. As such, it is necessary to repeat it more times than the rest in order to give a fair chance and opportunity for the players to have the possibility to win by it. Essentially, it is also common that by using the paper item in the game, it is likely that there will be many ties than wins. As a result, it is necessary to play paper more times in order to reduce the effect of the ties in the game . To play effectively, paper should be appointed half a portion of the entire playtime. Scissors should be appointed a quarter portion of the entire playtime. Rock should also have a quarter of the entire playtime. In total, this will make up time portions, which add up to one, which imply a full playtime. The rock is expected to have far much effect and impact in the game. As such, in order to mitigate possible adverse effects of rock, it is well necessary to ensure it is played less. This brings about equilibrium in the entire play matrix.
Question 2
If two people approach one another in a sidewalk, they face different directions. Say if one faces west, the other faces west. These people in order to pass one another without colliding, they should make their choices wisely . For instance, they would both choose to walk each on their right hand side or left hand side. When heading to different directions, the right hand side of each is the opposite of the other. This also applies for the left hand side. If they each follow similar sides while approaching one another, then they will pass one another without any problems. In this regard, both of them will receive a payoff of one. However, if they make opposite choices, then they will clash. Therefore, they will experience a problem when passing one another. They will probably squeeze or experience a head on collision. This way, they get a payoff of zero. If the two do not collide, they both get a payoff of one. However, when they do they both get a payoff of zero. This implies that there is a 0.5 probability of each getting ether payoff. This is because they choose either the right or left in which case if they choose the same side then each of them has a payoff of one. If not, so that they choose opposite directions, then the payoff is zero.
When we find the probability of either payoff, it is clear that it is a 0.5 probability. The total pay off for both when they make the same choice will be two since adding the payoffs totals to two. As such, it is clear that considering the mixed equilibrium, we find that it is indeed half as good for either player in the sidewalk game within the mixed equilibrium just as it is for the pure equilibriums. This is because there are half chances in the equilibriums that the players will get the same payoffs. However, it is clear that the choices made will be influenced by certain factors that the people should consider before they come near to each other. Each of them should determine the possible side that the other will follow and hence make his or her choice wisely if only to avoid a possible problem when passing one another . In each of the matrix, the chances prove to be 0.5 of falling into either case. Nevertheless, the pay offs are half as good in each of the three notable cases and one has an advantage over the other.
References
Brandon, C. (2009). Game Theory, Predicatibility and Route Selection. Infantry, Vol. 98, No. 3 , pp 3-53.
