Ridere, ludere, hoc est vivere.

Monday, October 15, 2018

Notes on games with mixed strategies


In an earlier post regarding theory of simultaneous move games, I concluded with an example of a game between two tennis players that did not demonstrate a Nash equilibrium between its two pure strategies. Sam Hillier: Consulting Philosopher more recently elaborated on the topic with an excellent post on mixed strategies. Whereas I had approached the question of an equilibrium for a single tennis shot and concluded that none existed, a tennis match of course includes many shots, so players have an opportunity to invoke a weighted mix of shots and defenses between the two options.

To recap, the two-player zero-sum simultaneous-move game represents a tennis volley in which one player - the defender - decides whether to defend a shot down the line (D) or cross court (C) at the same time the opponent - the shooter - decides which type of shot to attempt. The payoff is depicted as the probability of the shot scoring a point in a 2x2 matrix cross-referencing each player's decision with the other's. The shooter seeks to maximize that result; the defender wants to minimize it.

Sam stipulated that in a game of mixed strategies between C and D, “the server will shoot C 30% of the time … while the defender will defend against C 40% of the time,” but didn't explain where those mixed strategy results came from. There’s a little arithmetic in figuring that out that arises from approaching the problem by trying to “minimize the maximum loss.” 

Suppose p represents the frequency with which the shooter will select C. The shooter wants to maximize the worse case score regardless of the defender’s choice. If the defender chooses C, then the shooter’s expected value of the result will be

EC = 0.2p + 0.8(1-p)

If the defender chooses D, the shooter’s expected result is

ED = 0.9p + 0.5(1-p)

To avoid the defender exploiting one better result over another, the shooter will find the frequency p that gives the same result regardless of the chosen defense by setting the expected values equal to each other:

EC = ED

0.2p + 0.8(1-p) = 0.9p + 0.5(1-p)

p = (0.8 – 0.5) / (0.9 – 0.5 – 0.2 + 0.8)

= 0.3/1.0

= 30%

This result means that if the shooter chooses C 30% of the time, their expected score will be

EC = 0.2(0.3) + 0.8(1-0.3) = 0.62
or
ED = 0.9(0.3) + 0.5(1-0.3) = 0.62

regardless of the defender’s choice. If the shooter chooses C less frequently (i.e. p < 30%), the defender can always choose D and cause a lower shooter score. Similarly, if the shooter chooses C more frequently, the defender can always choose C and again cause a lower shooter score. So, the shooter’s best mixed strategy is to choose C 30% of the time.

Similar analysis – whereby the defender attempts to minimize the maximum possible shooter score – results in the defender choosing C

q = (0.9 – 0.5) / (0.8 – 0.5 – 0.2 + 0.9)

= 0.4/1.0

= 40%
of the time.

Sam also discusses the idea of Evolutionary Game Theory, which interests me from the standpoint that superior strategies can emerge in a population of game players without necessarily invoking explicit analysis. In fact, most games are sufficiently complex that exhaustive analysis is not possible, and the evolutionary strategies that emerge may demonstrate consistent success without a complete understanding of why they succeed. I expect Dr. Wictz and I will discuss that topic at some point in the future as well.

Monday, October 8, 2018

Spies vs spies: First impression of Cold Warrior

I have long felt that the Cold War represents a rich thematic opportunity for board games. Yet I can identify only a handful of games set in that NATO-Soviet contest of diplomacy, espionage, and brinkmanship that rank in the top 1000 of boardgamegeek.* Into this cloak-and-dagger arena the new designer Wes Crawford introduces Cold Warrior (artist Jimmy Malone, published on Game Crafter).

Monday, October 1, 2018

Quick draw: First impression of Death Pit Duels

In the world of head-to-head two-player battle card games, I'd be hard pressed to name a more distilled, purified entry than Death Pit Duels (designer Bryan Johnson, publisher Frost Forge on Game Crafter).

Monday, September 24, 2018

Notes on simultaneous-move games, and an exploration of the Stag Hunt

Some time ago, the design team of Dr. Wictz and I started discussing the book Games of Strategy by Dixit, Skeath, and Riley.  I wrote on a couple of topics:
In this post, I'd like to address simultaneous-move games with a specific focus on pure discrete strategies. (We recorded our discussion on this topic in April of last year.)  I recall such games represented in my earliest readings on game theory in the form of a decision-payoff matrix.  In a two-player game in which each player makes a single decision from among a finite number of choices, without knowledge of the other player's decision, the decision-payoff matrix labels the rows with one player's options and the columns with the other player's options.  The corresponding cell for a given combination of decisions yields the payoff to both players.

Monday, September 17, 2018

Magnificent feedback

Playtesting is crucial to any successful design, but the tricky part has often been which feedback to accept and which to ignore. Keith Ferguson and I really thought we had "Magnificent Marvels" nailed down when we pitched it at Origins and eventually signed it with Hexagram 63. The publisher identified some modifications for us to explore, so Keith tested some changes out at WashingCon and again at The Island Games, our friendly local game store. The changes that Hexagram 63 requested seem to work well, but some other feedback that Keith received surprised us somewhat. We have to look hard at where to make changes and where to stick with our original design.

Monday, September 10, 2018

Bidding and game theory

I have been thinking about the game theory construct for the share auction in Chicago Express (designer Harry Wu, publisher Queen). The question isn't only one of absolute valuation but also one of the interactive decision-making in the auction. That thought led to consideration of the auction as a game-theory problem.

Monday, September 3, 2018

Magnificent spreadsheets

There was a bit of a comparative discussion on Twitter among a few game designers about the use of spreadsheets. For my part, I find them useful in maintaining balance in a game's economy, in the relative values of different components of the game. In "Magnificent Marvels," Keith and I recognized the need to be sure that the different components with widely varied point values would need appropriately balanced building costs, and we put together a spreadsheet to try to manage that.

Monday, August 27, 2018

No end in sight

My friends and I played Axis & Allies: 1914 recently, and while I had fun, I was disappointed and irked about a fundamental design flaw in the game end conditions. The rules require one side to capture two opponents' capitals, of which one must be Paris or London (for a Central Powers victory) or Berlin (for an Allied victory). After five turns and eight hours, we had reached something of a stalemate - or at least a realization that the end of the game was still a long way off.