(I'm a CS student and have not learnt about Game Theory at Uni yet so my knowledge on this field is quite limited)
Imagine a game with 5 different stages, each stage giving a score of 1-10. At the end of the game, we get a score. There are 2 ways to get the final score:
- Sum up the scores from the 5 stages
- Multiply the scores from the 5 stages
As we play the game, we have to divide our efforts and get different scores for the stages. The 10 stages are all different and the player's ability to get a high score is different from stage to stage. Additionally, the stages can be either dependent or independent on each other:
- Independent: The performance of each stage has 0 effect on the performance on other stages. The skills required for every stage is also different.
- Dependent: Getting a high score on a stage makes the next stage easier or allows the player to get a higher score on the next stage. This can happen by applying a multiplier (+10% score if previous stage got >60%) or giving the player an advantage (in car racing, some racers are placed further in front, giving them an advantage and their probability of getting a better score is increased).
Example 1 - independent stages:
- Player 1 gets the scores: 5, 6, 7, 8, 9. Sum = 53; Product = 15120
- Player 2 gets the scores: 6, 6, 8, 8, 10. Sum = 38; Product = 23040
- P1 wins if scores are summed; P2 wins if scores are multiplied
Example 2 - dependent stages with multipliers:
- Score is multiplied by 10% if previous stage got >=60%
- P1 gets: 6, 7, 5, 6, 7. Sum = 6+7.1+5.1+6+7.1 = 31.3; Product = 6*7.1*5.1*6*7.1 = 9252.7
- P2 gets: 4, 9, 4, 6, 9. Sum = 4+9+4.1+6+9.1 = 32.2 Product = 4*10*4.1*6*10.1 = 8058.9
- P1 wins if stage scores are multiplied; P2 wins if stage scores are summed.
Example 3 - dependent stages with advantages:
- P1 and P2 are about equally skilled on average. The winner of a stage gets an advantage with gives them a 70% chance of getting a higher score the next stage (as opposed to 50%).
- P1's expected scores without an advantage: 6, 7, 5, 7, 6
- P2's expected scores without an advantage: 3, 9, 8, 4, 7
- Winner of the 5 stages without an advantage: P1, P2, P2, P1, P2
- Assuming the advantage gives a player +1 score
- P1's expected scores: 6, 8*0.7 + 7*0.3, 5*0.7 + 6*0.3, 7*0.7 + 8*0.3, 7*0.7 + 6*0.3; Sum = 33; Product = 11972.9
- P2's expected scores: 3, 9*0.7 + 10*0.3, 9*0.7 + 8*0.3, 5*0.7 + 5*0.3, 7*0.7 + 8*0.3; Sum = 33.3; Product = 8858.5
- P2 wins if scores are multiplied; P1 wins if scores are summed
In example 2 and 3, P1 spreads his effort out more evenly, while P2 focuses on some and does poorly on others.
Questions: Considering both dependent and independent stages:
- As the player, what are the differences in strategies to maximize the final score? (Flat and even vs Bumpy score line)
- As the game organizer, how can we choose between the 2 methods (depending on how we want the player to distribute their efforts)?