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Nash Equilibrium and Replicator Dynamics

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$begingroup$

Consider the matrix

$$ A = beginbmatrix 0 & 10 & 1 \ 10 & 0 & 1 \ 1 & 1 & 1 endbmatrix $$
i) Find the two Nash Equilibria.

ii) Consider the replicator dynamics along the invariant line $x_1=x_2$ to deduce the structure in the figure shown below.

$underlinetextbfMY WORK$

For some $x_1,x_2,w_1,w_2 in [0,1]$; where $textbfx= beginbmatrix x_1 \ 1-x_1 endbmatrix$ and $textbfw= beginbmatrix w_1 \ 1-w_1 endbmatrix$ we have:
$$textbfxcdot(Atextbfx) = beginbmatrix x_1 \ x_2 \ 1-x_1-x_2 endbmatrix beginbmatrix 0 & 10 & 1 \ 10 & 0 & 1 \ 1 & 1 & 1 endbmatrix beginbmatrix x_1 \ x_2 \ 1-x_1-x_2 endbmatrix $$
$$ =beginbmatrix x_1 \ x_2 \ 1-x_1-x_2 endbmatrix beginbmatrix -x_1+9x_2+1 \9x_1-x_2+1 \ 1 endbmatrix = 1+18x_1x_2-x_1^2-x_2^2 $$

and

$$textbfwcdot(Atextbfx) = beginbmatrix w_1 \ w_2 \ 1-w_1-w_2 endbmatrix beginbmatrix 0 & 10 & 1 \ 10 & 0 & 1 \ 1 & 1 & 1 endbmatrix beginbmatrix x_1 \ x_2 \ 1-x_1-x_2 endbmatrix $$
$$ =beginbmatrix w_1 \ w_2 \ 1-w_1-w_2 endbmatrix beginbmatrix -x_1+9x_2+1 \9x_1-x_2+1 \ 1 endbmatrix = 1 -w_1x_1 + 9w_2x_1 + 9w_1x_2 - w_2x_2 $$
not sure how to proceed.
Thank you for all help.

dynamical-systems game-theory nash-equilibrium evolutionary-game-theory

share|cite|improve this question

asked Mar 29 at 0:59

elcharlosmasterelcharlosmaster

2810

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  For the second question I recommend you to write down replicator equations in your question. This would be easier for others to answer your question if all relevant information is presented here.
  $endgroup$
  – Evgeny
  Mar 30 at 14:41
  

  
  
  
  

add a comment | 

0

$begingroup$

Consider the matrix

$$ A = beginbmatrix 0 & 10 & 1 \ 10 & 0 & 1 \ 1 & 1 & 1 endbmatrix $$
i) Find the two Nash Equilibria.

ii) Consider the replicator dynamics along the invariant line $x_1=x_2$ to deduce the structure in the figure shown below.

$underlinetextbfMY WORK$

For some $x_1,x_2,w_1,w_2 in [0,1]$; where $textbfx= beginbmatrix x_1 \ 1-x_1 endbmatrix$ and $textbfw= beginbmatrix w_1 \ 1-w_1 endbmatrix$ we have:
$$textbfxcdot(Atextbfx) = beginbmatrix x_1 \ x_2 \ 1-x_1-x_2 endbmatrix beginbmatrix 0 & 10 & 1 \ 10 & 0 & 1 \ 1 & 1 & 1 endbmatrix beginbmatrix x_1 \ x_2 \ 1-x_1-x_2 endbmatrix $$
$$ =beginbmatrix x_1 \ x_2 \ 1-x_1-x_2 endbmatrix beginbmatrix -x_1+9x_2+1 \9x_1-x_2+1 \ 1 endbmatrix = 1+18x_1x_2-x_1^2-x_2^2 $$

and

$$textbfwcdot(Atextbfx) = beginbmatrix w_1 \ w_2 \ 1-w_1-w_2 endbmatrix beginbmatrix 0 & 10 & 1 \ 10 & 0 & 1 \ 1 & 1 & 1 endbmatrix beginbmatrix x_1 \ x_2 \ 1-x_1-x_2 endbmatrix $$
$$ =beginbmatrix w_1 \ w_2 \ 1-w_1-w_2 endbmatrix beginbmatrix -x_1+9x_2+1 \9x_1-x_2+1 \ 1 endbmatrix = 1 -w_1x_1 + 9w_2x_1 + 9w_1x_2 - w_2x_2 $$
not sure how to proceed.
Thank you for all help.

dynamical-systems game-theory nash-equilibrium evolutionary-game-theory

share|cite|improve this question

asked Mar 29 at 0:59

elcharlosmasterelcharlosmaster

2810

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  For the second question I recommend you to write down replicator equations in your question. This would be easier for others to answer your question if all relevant information is presented here.
  $endgroup$
  – Evgeny
  Mar 30 at 14:41
  

  
  
  
  

add a comment | 

$begingroup$

Consider the matrix

$$ A = beginbmatrix 0 & 10 & 1 \ 10 & 0 & 1 \ 1 & 1 & 1 endbmatrix $$
i) Find the two Nash Equilibria.

ii) Consider the replicator dynamics along the invariant line $x_1=x_2$ to deduce the structure in the figure shown below.

$underlinetextbfMY WORK$

For some $x_1,x_2,w_1,w_2 in [0,1]$; where $textbfx= beginbmatrix x_1 \ 1-x_1 endbmatrix$ and $textbfw= beginbmatrix w_1 \ 1-w_1 endbmatrix$ we have:
$$textbfxcdot(Atextbfx) = beginbmatrix x_1 \ x_2 \ 1-x_1-x_2 endbmatrix beginbmatrix 0 & 10 & 1 \ 10 & 0 & 1 \ 1 & 1 & 1 endbmatrix beginbmatrix x_1 \ x_2 \ 1-x_1-x_2 endbmatrix $$
$$ =beginbmatrix x_1 \ x_2 \ 1-x_1-x_2 endbmatrix beginbmatrix -x_1+9x_2+1 \9x_1-x_2+1 \ 1 endbmatrix = 1+18x_1x_2-x_1^2-x_2^2 $$

and

$$textbfwcdot(Atextbfx) = beginbmatrix w_1 \ w_2 \ 1-w_1-w_2 endbmatrix beginbmatrix 0 & 10 & 1 \ 10 & 0 & 1 \ 1 & 1 & 1 endbmatrix beginbmatrix x_1 \ x_2 \ 1-x_1-x_2 endbmatrix $$
$$ =beginbmatrix w_1 \ w_2 \ 1-w_1-w_2 endbmatrix beginbmatrix -x_1+9x_2+1 \9x_1-x_2+1 \ 1 endbmatrix = 1 -w_1x_1 + 9w_2x_1 + 9w_1x_2 - w_2x_2 $$
not sure how to proceed.
Thank you for all help.

dynamical-systems game-theory nash-equilibrium evolutionary-game-theory

share|cite|improve this question

asked Mar 29 at 0:59

elcharlosmasterelcharlosmaster

2810

$endgroup$

share|cite|improve this question

asked Mar 29 at 0:59

elcharlosmasterelcharlosmaster

2810

share|cite|improve this question

asked Mar 29 at 0:59

elcharlosmasterelcharlosmaster

2810

asked Mar 29 at 0:59

elcharlosmasterelcharlosmaster

2810

asked Mar 29 at 0:59

elcharlosmasterelcharlosmaster

2810