Geoff Thomas
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In 2004, Friedemann Friese released the game "Power Grid". The aim of this game is to supply the most cities with power by building power plants, buying raw materials, connecting cities and powering them using power plants. Since then, several expansions have arrived bringing new maps, rules and power plants, but the core mechanics remain the same. In this review, I aim to provide some critical analysis of the gameplay of Power Grid.

Power Grid is an economic game built around the central currency of “Electros”. While the core aim of the game is to power as many cities as possible at the end of the game, in order to achieve this, players must spend their electros to expand their network. Ultimately, players compete to use their electros most efficiently although there are other considerations as well.

When a game is focused on efficiency, critics will often label the game as “mathsy” and this generally the most common complaint of Power Grid. So while there is mental arithmetic involved, there are other elements to Power Grid than purely maximising your economic engine, and while mathematical skills may be advantageous, they are by no means necessary.

We can see this most in the buying of cities. The action of buying cities parallels the 1930 mathematical problem, "The Travelling Salesman Problem". Of course, there are many differences between the two, the key problem being that there are other players and this brings challenges not thought of in the pure mathematical problem. Because other players are building cities, the topology of the city network is changing every round and so it’s not possible to just calculate the smallest network at the start of the game. Your starting city and that of your other players is very much the skill involved in the building of cities. Going for the cheapest position may not always be the best solution as there is likely to be increased competition. You also have to be wary of getting boxed in during the earlier stages of the game so that the only possible expansion is prohibitely expensive. The changing nature of the game prevents long-term formulation of the cheapest solution. However, generally in an individual round, players will want to find the cheapest solutions.

Optimisation is apparent elsewhere in Power Grid, but again, the other players in the game have a significant impact. As well as building cities you also require money to buy power plants and resources for them. However, other players will want to bid on power plants which may drive the price up, other players may purchase resources before you, again driving the price up. This competition will mean the most efficient strategy is constantly changing. This is why the game is not purely about maximising the expected returns, because other players have impact on the game and the imperfect, irrational nature of their play will result in the mathematical perfect play not always being the most effective. Power Grid artificially prevents players from making long term, deterministic plans. It does this by randomising the Power Plant deck (in all but one of the expansion maps, China). This is the only random element in Power Grid, but they are intertwined with the other uses of electros that they play a significant effect.

Power Plants have two interactions with the other elements of the game. They determine which resources players will need in order to power cities affecting the availability and price of resources. They also determine how many cities players are able to power, affecting both the players income and the need to build cities. So while this is a random element to the game, it prevents Power Grid from being purely a theoretical exercise.

So while this randomness does prevent long-term deterministic planning, players must still think ahead and not just on the immediate turn especially as the game progresses. Players must consider how they are going to power sufficient cities at the game end and buy power plants accordingly, often several turns ahead due to the buying limit, otherwise they may find themselves unable to power enough cities to win. This also means they need an understanding of when they think the game must end. So while players can’t know exactly what they are going to do, they must still have to have a rough idea.

Long term planning is also apparent due to the clever mechanic to determine the player order each turn. Some rounds in Power Grid operate in reverse order and because of this, players may intentionally fall behind in the number of cities in order to gain an earlier position in the turn order. Players later in the turn order get cheaper resources and first opportunity to build cities. But they’ll be getting less income and be further from winning. Some people think of this as a catch-up mechanic but more experienced players will understand the importance of manipulating their position in the turn order. Players need to balance sitting back to exploit the cheaper resources and building, building cities to make more money and when to get ahead to win.

All of this means Power Grid is a game high in player interaction. Players need not only think about themselves, but also what their opponents are doing. Unfortunately, Power Grid clouds this by allowing players to hide the amount of electros they have. This mechanic goes against all the open information available to player and feels artificial. You can understand why Power Grid did this, not only are electros used as a tie-breaker, but knowing how much everyone has could bog players down with over-thinking but it just seems out of place. What is particularly odd is that all the use and gain of electros is open and it would certainly be possible for players to calculate how much everyone so the game relies on the large number of actions for players to forget and to introduce this uncertainty in the state of other players. This does as some suspense to the game as players will be unaware how well their opponents are doing but it causes players to make suboptimal moves due to the uncertainty of the game state. Some may think this is a good thing but it takes away from the “pure” game that Power Grid is.

Another complaint of Power Grid is that repeat play does not garner enough differences between games. While Power Grid does offer multiple map expansions, these generally are limited to different city topologies and differing availability of Power Plants and resources. But really, multiple games will generate different player interactions. The amount of decisions in Power Grid and with a random Power Grid deck, no two play throughs should be the same. Unfortunately though, you are likely to see repetitive starts to the game as the initial power plants are always the same (some expansions do change this).

Overall, while there may be these minor problems in the game, it can’t detract from the great gameplay of Power Grid. If you are willing to look past the arithmetic there lies a great economic game full of player interaction that allows long-term thinking without getting bogged down in precision with important short-term decisions.
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I think traveling salesman problem is common to rail games, but it's a poor fit for Power Grid. In the crayon rail games, you are trying to get to places where particular goods are bought or sold. Transamerica is also a network game where you need to get to five particular places. In Power Grid you care about the number of cities you serve, but a city is a city, so the concept of solving the network is mushed down.
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Steven Backues
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Do you really think that the randomness in the power plant draws is necessary to prevent deterministic planning? I haven't tried the game without it, but I would think that the interaction inherent in the auction itself would be enough to prevent that. If you had some optimal path planned out, any other player could easily destroy it by bidding up the plant that you wanted.
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Geoff Thomas
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Power Grid is actually a "Minimum Spanning Tree" but the Travelling Salesman is a better known problem and the two are very similar, the only different being the spanning tree allows a tree to form rather than a single connecting line (which is rare for Power Grid).

It's precisely because the cities are just cities in Power Grid that it fits this problem (although the tree is hugley complex due to the ability to "jump over" cities although most can be ignored since they are so costly). The travelling salesman (and minimum spanning tree) is about visiting every city, in any order as cheaply as possible. Rail games are not about linking places as efficiently as possible (and in delivery games such as Steam, almost as inefficiently as possible). I haven't played Transamerica so I don't know how that relates, but it is certainly the case in Steam/Age of Steam/Railroad Tycoon and less so in 18XX.

Elendil wrote:
Do you really think that the randomness in the power plant draws is necessary to prevent deterministic planning? I haven't tried the game without it, but I would think that the interaction inherent in the auction itself would be enough to prevent that. If you had some optimal path planned out, any other player could easily destroy it by bidding up the plant that you wanted.


Mathematically, yes. With a fixed order Power Plant deck, the game has Perfect Information and you can determine every possible combination of moves that every player will make (including how much they will bid at auction). It's a very complicated problem, more so than Chess but it's certainly mathematically solveable with a fixed Power Plant deck.

Of course, in real play, full deterministic gameplay is unachievable. I've played the China scenario (which has fixed ordered Power Plants) and the game really isn't very different. Mostly because none of us knew what each Power Plant was so we didn't know what was coming to plan for it. However, I imagine those who did know the Power Plant deck would know how soon better plants were coming out, when a particular resource is likely to be in high demand that would cause them to alter their bids.

I hoped I've explained that well enough...
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AdmiralGT wrote:
Mathematically, yes. With a fixed order Power Plant deck, the game has Perfect Information

Also note that unknown random cards are removed from the game unless playing with 5 or 6 players.
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russ wrote:
AdmiralGT wrote:
Mathematically, yes. With a fixed order Power Plant deck, the game has Perfect Information

Also note that unknown random cards are removed from the game unless playing with 5 or 6 players.


My apologies, you are indeed correct that this is only applicable to 5/6 player.
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AdmiralGT wrote:
Elendil wrote:
Do you really think that the randomness in the power plant draws is necessary to prevent deterministic planning? I haven't tried the game without it, but I would think that the interaction inherent in the auction itself would be enough to prevent that. If you had some optimal path planned out, any other player could easily destroy it by bidding up the plant that you wanted.


Mathematically, yes. With a fixed order Power Plant deck, the game has Perfect Information and you can determine every possible combination of moves that every player will make (including how much they will bid at auction). It's a very complicated problem, more so than Chess but it's certainly mathematically solveable with a fixed Power Plant deck.

Of course, in real play, full deterministic gameplay is unachievable. I've played the China scenario (which has fixed ordered Power Plants) and the game really isn't very different. Mostly because none of us knew what each Power Plant was so we didn't know what was coming to plan for it. However, I imagine those who did know the Power Plant deck would know how soon better plants were coming out, when a particular resource is likely to be in high demand that would cause them to alter their bids.

I hoped I've explained that well enough...


I've been thinking a bit more about this, but I still don't really think I agree with your point. Let's assume that the power plant order was fixed and known, with nothing removed, and money was open. Yes, this would make it Perfect Information, but I don't think it would mean that you could therefore plot a perfect sequence of plays, even theoretically, unless you were only playing with 2 players.

You say that in that case "you can determine every possible combination of moves that every player will make (including how much they will bid at auction)." But you can't determine what moves players will make, only what moves they should make, in the sense of optimal moves for advancing their own position.

In a 2-player game, that's enough, because if your opponent makes a suboptimal move, all the better for you. But as soon as you add a third player, this no longer holds. A player could make a move that is both suboptimal for them and suboptimal for you, benefiting the third player. For example, if one player bids more on a particular power plant than it is actually worth to him and it is a plant that you wanted, then both you and he suffer, and the third player thereby benefits.

Moreover, one doesn't even need to invoke mistakes to see this effect. If one player (say, the start player) did somehow have an always-winning strategy, then it would be incumbent upon the other players to do something, anything, to stop that. For example, they should form an alliance, in which case the asymmetry of 1 vs 2 should allow them to defeat the first player even while making what might otherwise be suboptimal moves. I have seen people collaborate in the last moves of the game to try to stop a clear leader from winning; if the game was more transparent, perhaps this would happen earlier on.

Going back to the original thought: I am not really sure that the random power plant draws are necessary to keep Power Grid from becoming a "theoretical exercise" or the players from making "long term, deterministic plans." I suspect that the auction itself is enough to have that effect. The random power plant draws probably do shorten the planning window a little bit, but I am not sure it is really significant. Instead, I would guess that their main effect is to add additional tension and an element of risk management to the auction.
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Elendil wrote:
I've been thinking a bit more about this, but I still don't really think I agree with your point.

FWIW also based on what I've read of combinatorial games, 2-player combinatorial games (e.g. chess, go, Hex, etc as well as a hypothetically perfect information 2-player Power Grid) of course have optimal strategies (whether or not our puny minds can discover them is another question), but indeed the situation becomes mathematically much murkier with more than 2 players, partly for the reasons you mention. (Basically in a 2-player game, if my opponent screws up, that doesn't hurt me, but in a multiplayer game, if one of my opponents screws up, that can also hurt me.) Plus there are issues of collusion/alliance which can complicate things, and probably other stuff I'm not remembering.
 
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Elendil wrote:
I've been thinking a bit more about this, but I still don't really think I agree with your point. Let's assume that the power plant order was fixed and known, with nothing removed, and money was open. Yes, this would make it Perfect Information, but I don't think it would mean that you could therefore plot a perfect sequence of plays, even theoretically, unless you were only playing with 2 players.

You say that in that case "you can determine every possible combination of moves that every player will make (including how much they will bid at auction)." But you can't determine what moves players will make, only what moves they should make, in the sense of optimal moves for advancing their own position.

In a 2-player game, that's enough, because if your opponent makes a suboptimal move, all the better for you. But as soon as you add a third player, this no longer holds. A player could make a move that is both suboptimal for them and suboptimal for you, benefiting the third player. For example, if one player bids more on a particular power plant than it is actually worth to him and it is a plant that you wanted, then both you and he suffer, and the third player thereby benefits.

Moreover, one doesn't even need to invoke mistakes to see this effect. If one player (say, the start player) did somehow have an always-winning strategy, then it would be incumbent upon the other players to do something, anything, to stop that. For example, they should form an alliance, in which case the asymmetry of 1 vs 2 should allow them to defeat the first player even while making what might otherwise be suboptimal moves. I have seen people collaborate in the last moves of the game to try to stop a clear leader from winning; if the game was more transparent, perhaps this would happen earlier on.


There are 2 different concepts that I want to discuss here.

Firstly is that of "Perfect Information". By its very definition, if a game has perfect information (ie. the entire game state is known and every move is open), you are able to calculate every possible outcome in the game.

The second concept is that of a Solved Game. While the article only talks about 2-player games, the principles apply to multiplayer games. I'm not arguing that a know Plant deck means Power Grid is a Strong solveable game and that you can always plot a path to victory, but you can always plan your "best" result.

I've come up with a rather convoluted situation (using this to demonstrate how the fixed order Power Plant deck can lead to deterministic play. Lets say we have 3 players, A, B and C. Each of them can currently power 11 cities (having built 15) through 2 Power Plants and then have 160, 170 and 180 electros left to buy a replacement 3rd plant.

We have plants 36 to 42 available, with 44, 46 and 50 in the box. It will cost players 50 electros to build 2 cities, and 80 to build 3. 3 Coal cost 24 electros, 3 Oil 24 electros and 3 Garbage 12 electros. Players have enough resources to power all but the new Power Plant. The player order is currently A, B, C.

In the example of a known Power Plant deck (ascending order), Player C knows that Power Plant 46 will appear in 2 auctions time. Because of this, Player C can bid 36 for 36 or 48 for 38 and know that they will win. Either by winning those auctions or buying 46 for face value. (Paying either 70 for 46 + 24 (oil) while his opponents spend more than 60 - 48 + 12 (garbage) + bid over or 36 + 24 (coal) + bid over or by paying between 50 and 60 for another plant + resources).

Now suppose that the order of the last 3 Power Plants is not known. Now Player C has 2 options, bid 1 more on 36 and 38 hoping that 46 comes out, or bidding 91 (prevent A or B being able to afford to build and power 18 cities) or 81 (or 91 if B didn't bid on the first one, preventing the other of A and B from building 18 cities) and buying 50 at cost price (if they win neither of the first two). Of course, if they win the first auction at 91, A and B could collude to allow one of them to win. In this case, Player C can no longer determine what their best move is at the start of the turn.

We can apply this concept back through the game and so that without Perfect Information, the player has no idea of the best strategy. In this end-game solution (or even if the player knew 46 wasn't coming out), the solution is Strong solved. Because of the multiplayer nature of the game, I would doubt that it could be Strong solved at the start of the game.

Elendil wrote:
Going back to the original thought: I am not really sure that the random power plant draws are necessary to keep Power Grid from becoming a "theoretical exercise" or the players from making "long term, deterministic plans." I suspect that the auction itself is enough to have that effect. The random power plant draws probably do shorten the planning window a little bit, but I am not sure it is really significant. Instead, I would guess that their main effect is to add additional tension and an element of risk management to the auction.


I agree, the main effect is the add an element of risk management and uncertainty. Now while I doubt any human player would be able to calculate right from the start, my end-game scenario shows that a known order can be calculated (I wrote a couple of things down but it's certainly achievable working it out in your head). But I suspect players who knew the Power Plant deck would have a significant advantage, as I said in my review, particularly in knowing demand for resources and also when bigger Power Plants arrive (for example, I suspect knowing when Power Plant 20 is due would be advantageous). Because of this, I think the random nature of the Power Plant deck prevents people planning for these situations.

It was also mean poor replay value as people would quickly gather an understanding of how much everything is worth when.
 
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AdmiralGT wrote:
russ wrote:
AdmiralGT wrote:
Mathematically, yes. With a fixed order Power Plant deck, the game has Perfect Information

Also note that unknown random cards are removed from the game unless playing with 5 or 6 players.


My apologies, you are indeed correct that this is only applicable to 5/6 player.


On the contrary, the China scenario specifies which power plants that need to be pulled out if playing with less players... The whole point of China is that it IS predetermined and solvable...
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iamsum1gr8 wrote:
On the contrary, the China scenario specifies which power plants that need to be pulled out if playing with less players... The whole point of China is that it IS predetermined and solvable...

True true! It's been too long since I played China!

I actually thought from the context, though, that the comment wasn't talking specifically about China but about hypothetically eliminating the random plant order for any Power Grid scenario...
 
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I appreciate you taking the time to write out such a well-considered reply. I am sorry it has taken me so long to respond, but I have been thinking about the matter quite a bit.

I worked through the very clever example you constructed, and I agree, that in this case player C can calculate out and force a victory, whereas if the power plant draw was random he could not. So I agree, that's a good example of how the randomness of the power plant draw does limit one's ability to plan, perhaps even more than I was realizing. Of course, this is just one particular constructed example regarding a single turn, the last turn of the game. Most other situations, especially earlier in the game, would be more fluid, as I think (maybe?) you also agree.

Regarding "Perfect Information" and "Solved Game": I agree with what you say about "Perfect Information," but not about solved games. Specifically, you say "While the article only talks about 2-player games, the principles apply to multiplayer games." I don't agree that the principles apply to multiplayer games. There is a reason that the article only talks about 2-player games, and that is because multiplayer games can't generally be solved in the same sense that 2-player games can. I explained why above, as (briefly) did Russ; if you like you can also see this thread, where I (at that time doubting myself somewhat) put the question to the community as well.

This is the main point that I have been trying to make overall: multiplayer games are not solvable, and therefore you do not need to include random elements in order to keep a multiplayer game from being solvable. This is already excluded by default.

But maybe you also see this, because you also include some hedging statements, such as the very next sentence "I'm not arguing that a know Plant deck means Power Grid is a Strong solveable game and that you can always plot a path to victory, but you can always plan your "best" result." However, here I am not really clear on what you mean by a "best" result. And you also say later, "Because of the multiplayer nature of the game, I would doubt that it could be Strong solved at the start of the game." So we agree there, maybe...

I guess overall I am not quite sure where you stand, and that is why I am having difficulty responding. Do you think that multiplayer perfect information games are solvable? If so, I have to disagree. Certainly they can to some extent be analyzed, just as games without perfect information can also be analyzed (one can often calculate out a "most likely to win" play), but neither can be solved.

On the other hand, maybe you don't think (or don't think anymore) that multiplayer games are solvable, and are just saying that having perfect information allows more analysis. In that case, I certainly agree with you. Yes, if power plants were not random, you could do somewhat more calculation, and knowing the deck would be an advantage (even more than it already is). But there are fundamental limits to how much you could plan out because of the unpredictability of the other players. You could plan more, but not everything, nor even almost everything.

So like I said, I am not sure quite where you stand, and therefore I am not quite sure in what sense we disagree. It could be that our disagreement is only one of scale: how much of an effect would having fixed power plant draws have? If that is the case, then we might not be able to get much further, since that would ultimately be a matter of perception. (And at the moment, worse yet, a matter of speculation, as neither of us have really tried it).

So... I guess that turned out to be kind of a long and fuzzy post, the reason being that I am not quite sure what your position exactly is. But I wanted to write something since you had put the effort in and made that nice example, so hopefully this helps to clarify things somewhat anyhow.
 
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