Game Theory Nash Equilibrium - Iterative Closed Bag Exchange Donation Game
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I am making up a game for a Holiday party and became interested in checking if the game is somehow broken or if there is an optimal solution, which would make it less enjoyable. A Nash equilibrium may be a decent place to start with this, but I quickly realized I have very little understanding of game theory, especially when applied to iterative games.
Here is the game I have created:
Let's say two competitors exist - Astro and Pheonix - which are racing to get enough resources to fly to the moon.
The resources required to get to the moon are slightly different between the two teams (due to differences in rocket construction). As an example, let's say $R_{Astro} = 525$ and $R_{Pheonix} = 500$.
In order to create one $R$, a hydrazine block $H$ and one dinitrogen tetroxide block $T$ are required ($R=H+T$).
Each team is assigned only one of these resources (either $H$ or $T$).
The teams must trade resources in order to reach their goal. However, as the transaction occurs with a closed bag, the players can provide nothing in the exchange and potentially gain the other's resource without cost of their own.
The goals of each team are known to each other (so the team with a smaller number of resources needed may be careful as they know the other team will be less likely to give up resources); however, the number of resources which the team has on hand is not known to the other team.
I assume a decent way of calculating the number of resources given to each team should be based on the minimal amount of resources required for any company to get their goal ($R_{base} = 2times max(R_{Pheonix}, R_{Astro} = 1050$), plus a random number of resources based on this value (to provide some uncertainty of wealth for each team against each other).
For example, $random(R_{base},R_{base}+.05 times R_{base})$ or both Astro and Pheonix.
i.e. Astro has $1083 H$ and Pheonix has $1064 T$
I understand that this is an iterated, closed bag exchange, donation game as the Wikipedia explains
Here, $b$ is the value of the resource the team is seeking, while $c$ is the value of the resource for which the team initially starts.
There are two problems:
$b$ and $c$ change throughout the progression of the game- 0/0 (both give nothing) appears to be a Nash equilibrium - although that's only viewing $H+T$ as a score, not $R$, which requires equal parts of both.
In order to solve (2), I have included a mechanic to take away $2 H/T$ (H or T, of team's choice) every time both teams defect (give nothing).
Given these rules, is there an optimal strategy for this game?
game-theory nash-equilibrium
asked Dec 10 '18 at 17:12
chasechase
1326
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add a comment |
$begingroup$
I am making up a game for a Holiday party and became interested in checking if the game is somehow broken or if there is an optimal solution, which would make it less enjoyable. A Nash equilibrium may be a decent place to start with this, but I quickly realized I have very little understanding of game theory, especially when applied to iterative games.
Here is the game I have created:
Let's say two competitors exist - Astro and Pheonix - which are racing to get enough resources to fly to the moon.
The resources required to get to the moon are slightly different between the two teams (due to differences in rocket construction). As an example, let's say $R_{Astro} = 525$ and $R_{Pheonix} = 500$.
In order to create one $R$, a hydrazine block $H$ and one dinitrogen tetroxide block $T$ are required ($R=H+T$).
Each team is assigned only one of these resources (either $H$ or $T$).
The teams must trade resources in order to reach their goal. However, as the transaction occurs with a closed bag, the players can provide nothing in the exchange and potentially gain the other's resource without cost of their own.
The goals of each team are known to each other (so the team with a smaller number of resources needed may be careful as they know the other team will be less likely to give up resources); however, the number of resources which the team has on hand is not known to the other team.
I assume a decent way of calculating the number of resources given to each team should be based on the minimal amount of resources required for any company to get their goal ($R_{base} = 2times max(R_{Pheonix}, R_{Astro} = 1050$), plus a random number of resources based on this value (to provide some uncertainty of wealth for each team against each other).
For example, $random(R_{base},R_{base}+.05 times R_{base})$ or both Astro and Pheonix.
i.e. Astro has $1083 H$ and Pheonix has $1064 T$
I understand that this is an iterated, closed bag exchange, donation game as the Wikipedia explains
Here, $b$ is the value of the resource the team is seeking, while $c$ is the value of the resource for which the team initially starts.
There are two problems:
$b$ and $c$ change throughout the progression of the game- 0/0 (both give nothing) appears to be a Nash equilibrium - although that's only viewing $H+T$ as a score, not $R$, which requires equal parts of both.
In order to solve (2), I have included a mechanic to take away $2 H/T$ (H or T, of team's choice) every time both teams defect (give nothing).
Given these rules, is there an optimal strategy for this game?
game-theory nash-equilibrium
asked Dec 10 '18 at 17:12
chasechase
1326
$endgroup$
add a comment |
$begingroup$
I am making up a game for a Holiday party and became interested in checking if the game is somehow broken or if there is an optimal solution, which would make it less enjoyable. A Nash equilibrium may be a decent place to start with this, but I quickly realized I have very little understanding of game theory, especially when applied to iterative games.
Here is the game I have created:
Let's say two competitors exist - Astro and Pheonix - which are racing to get enough resources to fly to the moon.
The resources required to get to the moon are slightly different between the two teams (due to differences in rocket construction). As an example, let's say $R_{Astro} = 525$ and $R_{Pheonix} = 500$.
In order to create one $R$, a hydrazine block $H$ and one dinitrogen tetroxide block $T$ are required ($R=H+T$).
Each team is assigned only one of these resources (either $H$ or $T$).
The teams must trade resources in order to reach their goal. However, as the transaction occurs with a closed bag, the players can provide nothing in the exchange and potentially gain the other's resource without cost of their own.
The goals of each team are known to each other (so the team with a smaller number of resources needed may be careful as they know the other team will be less likely to give up resources); however, the number of resources which the team has on hand is not known to the other team.
I assume a decent way of calculating the number of resources given to each team should be based on the minimal amount of resources required for any company to get their goal ($R_{base} = 2times max(R_{Pheonix}, R_{Astro} = 1050$), plus a random number of resources based on this value (to provide some uncertainty of wealth for each team against each other).
For example, $random(R_{base},R_{base}+.05 times R_{base})$ or both Astro and Pheonix.
i.e. Astro has $1083 H$ and Pheonix has $1064 T$
I understand that this is an iterated, closed bag exchange, donation game as the Wikipedia explains
Here, $b$ is the value of the resource the team is seeking, while $c$ is the value of the resource for which the team initially starts.
There are two problems:
$b$ and $c$ change throughout the progression of the game- 0/0 (both give nothing) appears to be a Nash equilibrium - although that's only viewing $H+T$ as a score, not $R$, which requires equal parts of both.
In order to solve (2), I have included a mechanic to take away $2 H/T$ (H or T, of team's choice) every time both teams defect (give nothing).
Given these rules, is there an optimal strategy for this game?
game-theory nash-equilibrium
asked Dec 10 '18 at 17:12
chasechase
1326
$endgroup$
I am making up a game for a Holiday party and became interested in checking if the game is somehow broken or if there is an optimal solution, which would make it less enjoyable. A Nash equilibrium may be a decent place to start with this, but I quickly realized I have very little understanding of game theory, especially when applied to iterative games.
Here is the game I have created:
Let's say two competitors exist - Astro and Pheonix - which are racing to get enough resources to fly to the moon.
The resources required to get to the moon are slightly different between the two teams (due to differences in rocket construction). As an example, let's say $R_{Astro} = 525$ and $R_{Pheonix} = 500$.
In order to create one $R$, a hydrazine block $H$ and one dinitrogen tetroxide block $T$ are required ($R=H+T$).
Each team is assigned only one of these resources (either $H$ or $T$).
The teams must trade resources in order to reach their goal. However, as the transaction occurs with a closed bag, the players can provide nothing in the exchange and potentially gain the other's resource without cost of their own.
The goals of each team are known to each other (so the team with a smaller number of resources needed may be careful as they know the other team will be less likely to give up resources); however, the number of resources which the team has on hand is not known to the other team.
I assume a decent way of calculating the number of resources given to each team should be based on the minimal amount of resources required for any company to get their goal ($R_{base} = 2times max(R_{Pheonix}, R_{Astro} = 1050$), plus a random number of resources based on this value (to provide some uncertainty of wealth for each team against each other).
For example, $random(R_{base},R_{base}+.05 times R_{base})$ or both Astro and Pheonix.
i.e. Astro has $1083 H$ and Pheonix has $1064 T$
I understand that this is an iterated, closed bag exchange, donation game as the Wikipedia explains
Here, $b$ is the value of the resource the team is seeking, while $c$ is the value of the resource for which the team initially starts.
There are two problems:
$b$ and $c$ change throughout the progression of the game- 0/0 (both give nothing) appears to be a Nash equilibrium - although that's only viewing $H+T$ as a score, not $R$, which requires equal parts of both.
In order to solve (2), I have included a mechanic to take away $2 H/T$ (H or T, of team's choice) every time both teams defect (give nothing).
Given these rules, is there an optimal strategy for this game?
game-theory nash-equilibrium
game-theory nash-equilibrium
asked Dec 10 '18 at 17:12
chasechase
1326
asked Dec 10 '18 at 17:12
chasechase
1326
asked Dec 10 '18 at 17:12
chasechase
1326
asked Dec 10 '18 at 17:12
chasechase
1326
asked Dec 10 '18 at 17:12
chasechase
1326
1326
add a comment |
add a comment |
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