Simple summation, difficult inference. (game theory)

In summary, A user asked for help understanding how to substitute variables in summation signs, specifically in a proof for the payoff of a skew symmetric game. Another user provided a helpful explanation and example using sigma notation. The user was able to understand the concept after 10 minutes compared to the 3 hours it took to understand the proof.
  • #1
citrusvanilla
4
0
hi guys, new user, long time lurker.

the following simple proof is proposed with the highlighted summations. you do not need to know what the proof is of to answer my question (it is that the payoff for a skew symmetric game, rock paper scissors, is zero). i need help understanding how you can substitute variables in summation signs. specifically, how the sum of the term in row 1 from 1 to n is somehow equivalent to the sum in row 2 from i to n. this doesn't make any sense to me. how can something be summed from i to n?

fyi assume x_i = y_i for all i

http://i.imgur.com/s9cdo.jpg

http://i.imgur.com/s9cdo.jpg
 
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  • #2
The limits of the inside sigma can depend upon the variable of the outside.
Like if we have
1+(1+1)+(1+1+1)
we could write
[tex]\Sigma_{i=1}^3\Sigma_{j=i}^3 1[/tex]

if we have
x3+(x2+x3)+(x1+x2+x3)
we could write
[tex]\Sigma_{i=1}^3\Sigma_{j=i}^3 x^j[/tex]

to manipulate such we can focus on the possible value of each variable
if 0<i,j<n
if we prefer to work with i+j and i instead of j and i we have
0<i<n
i<i+j<n+i
 
  • #3
thanks lurf i really appreciate it. after staring at your answer for 10 minutes i understood it, this is compared to the 3 hours i took to not understand the proof.
 

1. What is simple summation in game theory?

Simple summation in game theory refers to the process of adding up the possible outcomes and their associated payoffs for each player in a game. It is a way to analyze the potential outcomes of a game and determine the best course of action for each player.

2. What is difficult inference in game theory?

Difficult inference in game theory is the process of using limited information and making educated guesses about the actions and strategies of other players in a game. It involves making inferences and predictions based on the available information and the behavior of other players.

3. How do simple summation and difficult inference relate in game theory?

Simple summation and difficult inference are both important tools in game theory. Simple summation helps to analyze the potential outcomes of a game, while difficult inference helps to predict the behavior of other players. Together, they can provide a more complete understanding of a game and inform decision-making.

4. What are some challenges associated with simple summation and difficult inference in game theory?

One challenge with simple summation is that it assumes all players are rational and will make decisions based on their own self-interest. This may not always be the case in real-world scenarios. Difficult inference can also be challenging because it relies on making assumptions and predictions about the behavior of others, which may not always be accurate.

5. How can game theory be applied in real-world situations?

Game theory has many applications in various fields such as economics, politics, and business. It can be used to analyze and predict the behavior of individuals and groups in competitive situations, and inform decision-making in complex scenarios. For example, game theory has been used to study negotiation strategies, market competition, and international relations.

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