user13829: Many world champion games are drawn, and people in the general public think that is disappointing. Not to debate that opinion here, but the question seems to be out there. So what would a rule change look like to get rid of most of the drawn games in tournaments? Especially those agreed upon rather than those forced. What has been proposed?
What consequences would come from giving zero points to both players who draw? Or to punish the party who proposes a draw by a reduced score? Draws can be forced, and the players could "agree to force" a draw, so it's not apparent to me how it would work out.

user703: You can incentivize any behavior by providing the appropriate reward function. It is hard to select proper rewards (and even harder to persuade other people to adopt them). One example would be to give a slight edge for a win: 1.25 for win, 0.5 for a draw. 
Another approach to make draws less likely is to start from a list of predefined very unbalanced positions. For example in checkers which had very high percentage of draws on the highest level, the game started not from the start but from a random position from a predefined list. 

user2415: The 3-1-0 system (3 for a win, 1 each for a draw) has been used but IMO it unfairly penalizes the player of Black. At strong GM level it's difficult to win with Black if the opponent only wants a draw.
My suggestion is to modify the 3-1-0 method by giving Black 1.5 for a draw. This is equivalent to changing the normal 1-0.5-0 system by only giving White 1/3 for a draw.
It wouldn't be at all suitable for Swisses but for the invitational all-by-all events I think it would be very positive. There would be much more onus on players with White to play for wins and risk-taking players would be rewarded. 

user4418: 
So what would a rule change look like to get rid of all the drawn
  games in tournaments?

It wouldn't look like anything to do with chess because the question is one which displays ignorance of the laws of chess.
There are many situations where a draw is automatic -
1) Stalemate
2) Insufficient mating material
3) Five fold repetition
4) 75 moves without a capture or pawn move.
All of these instantly end the game according to the latest FIDE Laws of chess and in these cases the result is a draw.

What consequences would come from giving zero points to both players
  who draw?

This shows tremendous disrespect to players who fight long hard games which end in draws because the players are evenly matched.

user18302: Possible solution for the World Championship match: always playing a tie-break before the classical phase.
In case, the main phase (12 classical games) is not drawn, the tie-break will be of no importance for selecting the world champion. Simply, the winner of the classical phase is always declared the world champion. Only if classical phase ends in a draw (6.0 – 6.0), the – previously played – tie-break is decisive.
The key idea is that a tie-break loser will have stronger incentives to play aggressively (offensively) than now. Already before the first classical game, he will be certain that without winning at least a single game out of 12, it is impossible for him to become the champion.
The incentives would be the strongest in the 12th game if after the 11 games the result is 5.5 – 5.5. To have championship chances, the tie-breaker loser will have to risk playing for a win.
The idea described here is a direct analogy to the solution proposed in the literature for soccer (see References below).  
References:
Carrillo, J. D. (2007). Penalty shoot-outs: Before or after extra-time? Journal of Sports Economics, 8, 505-518. (link)
Lenten, Liam JA, Jan Libich, and Petr Stehlík. "Policy Timing and Footballers' Incentives: Penalties Before or After Extra Time?." Journal of Sports Economics 14.6 (2013): 629-655. (link)

user12449: Tried systems are to award 3 points for a win and 1 for a draw, not being able to offer a draw before move 30, and only being able to draw with the consent of the arbiter.  There has been a long discussion about giving black 0.6 points versus white's 0.4, due the the advantage of the first move.  There is also Fischer's suggestion of not counting draws in matches.
I have suggested on splitting the point in a ratio derived by the difference in rating.  For people with <= 75 points difference, the point is split evenly.  75-150, the point is split 0.6-0.4, weaker and stronger player respectively.  There could also be a variance based upon which color was played.  This should give the stronger player more incentive to play for a win, however it also gives the weaker player more reward for playing for the draw. 

13829: <p>Many world champion games are drawn, and people in the general public think that is disappointing. Not to debate that opinion here, but the question seems to be out there. So what would a rule change look like to get rid of most of the drawn games in tournaments? Especially those agreed upon rather than those forced. What has been proposed?</p>

<p>What consequences would come from giving zero points to both players who draw? Or to punish the party who proposes a draw by a reduced score? Draws can be forced, and the players could "agree to force" a draw, so it's not apparent to me how it would work out.</p>
 703: <p>You can incentivize any behavior by providing the appropriate reward function. It is hard to select proper rewards (and even harder to persuade other people to adopt them). One example would be to give a slight edge for a win: 1.25 for win, 0.5 for a draw. </p>

<p>Another approach to make draws less likely is to start from a list of predefined very unbalanced positions. For example in checkers which had very high percentage of draws on the highest level, the game started not from the start but from a random position from a predefined list. </p>
 2415: <p>The 3-1-0 system (3 for a win, 1 each for a draw) has been used but IMO it unfairly penalizes the player of Black. At strong GM level it's difficult to win with Black if the opponent only wants a draw.</p>

<p>My suggestion is to modify the 3-1-0 method by giving Black 1.5 for a draw. This is equivalent to changing the normal 1-0.5-0 system by only giving White 1/3 for a draw.</p>

<p>It wouldn't be at all suitable for Swisses but for the invitational all-by-all events I think it would be very positive. There would be much more onus on players with White to play for wins and risk-taking players would be rewarded. </p>
 4418: <blockquote>
  <p>So what would a rule change look like to get rid of all the drawn
  games in tournaments?</p>
</blockquote>

<p>It wouldn't look like anything to do with chess because the question is one which displays ignorance of the laws of chess.</p>

<p>There are many situations where a draw is automatic -</p>

<p>1) Stalemate</p>

<p>2) Insufficient mating material</p>

<p>3) Five fold repetition</p>

<p>4) 75 moves without a capture or pawn move.</p>

<p>All of these instantly end the game according to the latest FIDE Laws of chess and in these cases the result is a draw.</p>

<blockquote>
  <p>What consequences would come from giving zero points to both players
  who draw?</p>
</blockquote>

<p>This shows tremendous disrespect to players who fight long hard games which end in draws because the players are evenly matched.</p>
 18302: <p>Possible solution for the World Championship match: always playing a tie-break before the classical phase.</p>

<p>In case, the main phase (12 classical games) is not drawn, the tie-break will be of no importance for selecting the world champion. Simply, the winner of the classical phase is always declared the world champion. Only if classical phase ends in a draw (6.0 – 6.0), the – previously played – tie-break is decisive.</p>

<p>The key idea is that a tie-break loser will have stronger incentives to play aggressively (offensively) than now. Already before the first classical game, he will be certain that without winning at least a single game out of 12, it is impossible for him to become the champion.</p>

<p>The incentives would be the strongest in the 12th game if after the 11 games the result is 5.5 – 5.5. To have championship chances, the tie-breaker loser will have to risk playing for a win.</p>

<p>The idea described here is a direct analogy to the solution proposed in the literature for soccer (see References below).  </p>

<p>References:</p>

<p>Carrillo, J. D. (2007). Penalty shoot-outs: Before or after extra-time? Journal of Sports Economics, 8, 505-518. (link)</p>

<p>Lenten, Liam JA, Jan Libich, and Petr Stehlík. "Policy Timing and Footballers' Incentives: Penalties Before or After Extra Time?." Journal of Sports Economics 14.6 (2013): 629-655. (link)</p>
 12449: <p>Tried systems are to award 3 points for a win and 1 for a draw, not being able to offer a draw before move 30, and only being able to draw with the consent of the arbiter.  There has been a long discussion about giving black 0.6 points versus white's 0.4, due the the advantage of the first move.  There is also Fischer's suggestion of not counting draws in matches.</p>

<p>I have suggested on splitting the point in a ratio derived by the difference in rating.  For people with &lt;= 75 points difference, the point is split evenly.  75-150, the point is split 0.6-0.4, weaker and stronger player respectively.  There could also be a variance based upon which color was played.  This should give the stronger player more incentive to play for a win, however it also gives the weaker player more reward for playing for the draw. </p>
