When to Buy: the Math in Imperial

Imperial is a 2006 game from Mac Gerdts that rethinks the classic Diplomacy. A layman—in this case, a true board game geek—would have difficulty not mistaking the maps incorporated in both games. What I like about Imperial is that it takes the too-interactive, too-chesslike Diplomacy, adds a realism-contributing economic aspect, and comes out with something that requires even more social and mental depth.

A critical component of the Imperial economy are “bonds”. Players seek to acquire bonds because they grant both points and influence within the game. For each of the six powers in the game, eight distinct bonds are available. Bonds feature a face value and interest value. For instance, the lowest-valued bond is the 1-bond, so called because its interest value is $1million. What this means is that for an initial investment (i.e. buying the bond) for $2m, you receive $1m payments each time a payout is to be had. This occurs indefinitely, so you have the possibility for making back your initial investment and more as the game goes on. When you’re rich enough, you might be able to afford the highest-priced bond, the 8-bond. As you could assume, its interest payment is $8m on a face value of $25m. Notice that the one-time interest payments as a fraction of the initial investment become smaller, the higher the bond’s face value. Viz. $1m/$2m = 50% while $8m/$25 = 32%.

Bond interest payouts not only provide the income stream described above, but also the points needed to win the game. At the game’s end, you can multiply the interest value of a bond by its power multiplier, a multiplier determined by how well a nation did in the game. Thus, you would like to be holding all of your high-valued bonds in the nations that did well and low-valued or, better, no bonds in the nations that did poorly.

However, the opportunities you have to buy bonds are limited and the cash available at that instant limited even more, so the question of what the “right” bond to buy is one that will cross your mind several times throughout a game. In particular, because the power multipliers will not be resolved until the game is already over, it’s important to forecast a nation’s finishing position to decide how much to invest. Just as investing too little is problematic because you miss out on multiplying more of your investment by those power multipliers, investing too much is problematic as it will stunt your ability to “grow” your money as the game goes on; in other words, what you spend this turn isn’t available next turn.

With that groundwork laid, it’s time to get to some pictures. First, a chart to capture what I have been discussing and provide some math that we’ll need.

ImperialPoints

Net payoff of bond purchase for varying multipliers.

This chart shows the net payoff from investing in varying bonds (the vertical axis). Of course, the net payoff is conditional on the power multiplier at game end (the horizontal axis). The net payoff is defined as the multiplier points less the original bond cost (face value). Thus, we see that the 8-bond paying off with the x5 multiplier yields 8×5=40 gross points. From this, we subtract the initial amount paid to buy the bond, 25, to get a net payoff of 15, the highest offered in the game.

What should also grab your attention, though, is that you can accomplish the same thing with the 7-bond (7×5–20=15). This begs the question: why buy the 8-bond. The answer is invariably “because the 7 has already been bought”.* As mentioned previously, it’s better to hold on to your money if you can. (You can keep it to add to your score at game’s end, so it never goes to waste.) Thus, in the perfect world, we absolutely make sure we have our hands on the 7-bond in x5 countries, the 5-bond in x4 nations, the 3-bond in x3 nations, and no higher than a 3-bond in any nations that fall short of x3. Anything else is gravy as long as you’re not taking a loss (red boxes).

Of course, the world’s not perfect and you’re never going to have your money in all the right places. There’s too many other players interested in seeing that you don’t. But it can pay to come close and know what you’re shooting for, although this will require taking some probabilities into account.

During a game in progress, you’ll only be able to discern game-end multipliers to a statistical certainty. The relative probabilities of each outcome dictate which bonds to shoot for. If you think x4 is likely with an outside shot of x3, then it pays most to aim for the 5-bond or the 4-bond.

Payoff graph

The progression of net payoffs for each bond

Since the probabilities will always be changing and the math is complicated to do in-game in the first place, it helps to generalize which bonds provide the best bang for the buck. The chart above attempts to capture this. At the start of the game, it assumes the following probabilties of a nation finishing in a given multiplier:

x0   0.00%
x1   5.00%
x2  25.00%
x3  33.33%
x4  20.00%
x5  16.66%

This suffices for the beginning of the game, but as the game progresses, nations will reach higher multipliers thereby nullifying the probability that they will reach a previous level—they’ve already reached it! For this, we make a simplifying assumption that the probabilities of reaching higher levels redistribute proportionally among the remaining possible outcomes. An example will help illustrate this. Once we reach x2, the chance of being at x1 is now 0. Thus, we redistribute the original 5% chance of being at x1 among the remaining outcomes in proportion to their original probabilities. Thus, the x2 outcome gains an extra (25%/95%)*5% chance, x3 adds (33.33%/95%)*5%, x4 goes up (20%/95%)*5%, and x5 increases (16.66%/95%)*5%. For each bond, we multiply the payoff at each level by the probability of realizing that payoff, and we generate the expected payoff for each bond.

As you would expect, as we can become more certain of reaching a given level, the expected net payoff increases. Thus, in the chart above, the right edge of each color band show the expected payoff assuming a given multiplier position. The dark blue is the result at the start of the game. Here we find that the 5-bond provides the best value for money as its blue bar reaches furthest right. The 5-bond falls about even with the 6-bond once we’ve reached the green region and sit in the x3 multiplier. Once we’re at x4, the 7-bond becomes the best and it maintains this positions even as the 8-bond evens up with it at the x5 region. This was precisely what we had found earlier.

Thus, we can see that the 5-bond is a very safe early bond to own in any country. It provides an optimal value at the x3 region, the optimum at x4, and a good return even at x5. However, once a country has attained x3. It’s best to focus on the 7-bond if possible. We also see the losses that can be sustained by investing in the 8-bond when it falls short of the x5 multiplier.

__________

* There are other gameplay factors at work here such as trying to take control of a nation. These effects are ignored in this analysis.

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