It's easy to look at a wargame and think that it's essentially "just" the combat system of an RPG. All the G without the R or the P. Similarly, it's easy to look at the original RPG (the fabled Chainmail supplement by Gygax and Arneson, I mean) and notice that the RPG was meant to be all the stuff but the combat system. It was the plot and the personality and the fantasy, but it still relied on a medieval wargame for combat. Wargaming today remains very structured by phases, or rounds, or turns. Roleplaying games today don't exactly have a concept of rounds or turns except during combat. What's interesting, though, is that wargaming has continued to evolve in parallel with RPG, and many of today's wargames have incorporated characters, leveling-up, transferable loot, shopping and trading, campaigns, and storytelling. These are two different approaches to essentially reach the same end.
Tabletop games usually have variable time. In fact, that's one of the reasons we play them. It's great fun to have the opportunity to see a problem that could be detrimental to achieving your goal, and then to pause the passage of [in-game] time, have a wee ponder about your options, follow the consequences of a possible decision to its logical conclusion, and then take action. You don't always get to do that in real life, and even when you can take time to make a decision, social engineering 101 ensures that the world's trying to convince you it's vital to act now, while the offer is still valid, and that the offer's only good while supplies last, so act now! Tabletop games are so much a thought exercise that one variant of chess requires players to clock in and clock out for each turn, just to prevent the game from taking days to complete.
As with other tabletop games, RPG and wargaming players tend to appreciate that time is variable in game. You can ponder your next move carefully, and commit to an action once you're comfortable with the probable outcome.
In an RPG, the stage of pondering possible consequences is, uniquely, often a group endeavour. It's a cooperative game, so what one player does often affects the other players, so in a way each player has several characters in play. Officially, you play only your own character. Unofficially, you're probably a friendly advisor (when your opinion is requested, anyway) for two or three other characters. It's not uncommon for an RPG group to be faced with a simple action that would take 6 seconds in the game world to perform, and to debate whether to take that action for an hour.
That's a difference by a factor of 600, and many (if not most) RPG players wouldn't trade that tradition of gross inefficiency and blatant lack of focus for the world.
Wargaming also has variable time, during play the factor is usually less than in an RPG. The difference between in-game action and Table Time decision is often a matter of a soldier (or a squad of soldiers) pulling a trigger or releasing a bow string and the player measuring distances, rolling dice, and referring to a chart in a book. It's usually a matter of 5 or 10 minutes, or up to 15 assuming a very patient opponent. A single turn taking an hour would be unheard of, and probably not tolerated.
This is an important difference between wargaming and RPG, though. In a wargame, everything happens during a turn. In an RPG, there are no turns except during combat.
A player turn is one important indication that a game is happening. Without turns, you often don't feel the momentum of a game. Some game systems use counters or imaginary resources that depreciate as the game goes on, but unless these are governed by a real world clock it's only by the passage of player turns that these indicators change state. There's no "time" in the game world without turns.
In an RPG, the Game Master must track time as it occurs in both turn-based mode and "roleplay mode".. The Game Master is expected to balance the need for the game to progress with the desire for players to discuss the problems their characters are facing, and then to push the players to make a decision. Personally, I track time using a deck of cards. When table talk has strayed far from character discussion, I sometimes ask a player whether their character is voicing their theory or not. If the character's thoughts are the same as the player's discussion, I draw a card to show that time has passed. After some number of cards, I can justify a random encounter or some in-game event to signify that game time is indeed passing.
In a wargame, everything happens within the structure of a turn. When one of your soldiers loots a treasure chest, that's their action. You roll a die or draw a card to find out what they've looted, and then move on with the game. When one of your squads is caught out by a trap, you roll a dice to learn the effect, and then move on. When a squad moves, you measure the distance, you move your miniatures.
That kind of regimented schedule doesn't generally allow for in-depth conversations or careful exploration. Some wargamers do play it that way, though. If you appoint a Game Master, your soldiers can explore a town or a dungeon or a space station and talk to NPCs or activate computers or examine puzzle boxes for hours. Make the wargame a cooperative game, and it's easy to RPG within a wargame, at least within the constraints of the wargame scenario (a wargame "requires" terrain and movement measurement, while an RPG tends to happen mostly in the mind's eye).
More commonly, wargames reserve the part of the game without turns to post-game activities. For example, after a successful mission you might have found a new weapon or new technology that you need to add to your army stats. Maybe you've earned currency you can spend at the local trading post. You might have lost a few soldiers along the way, so maybe you need to recruit new members into your team. There's often a lot of "solo RPG" after a wargame scenario, during which you read up on new rules, spend imaginary currency, roll on random tables, and so on. It happens after the turns have ended, but it's still done according to rules and according to events that took place in game. It's unquestionably part of the wargame experience, just as building or leveling-up a character is a part of the RPG experience.
Personally, the relief I feel when combat has ended and everyone is "out of initiative" is one of my favourite moments in an RPG. The turbulent transition from one game mode to another is something I enjoy, partly because it feels familiar and nostalgic because that's the way it's always been. I think it also mimics the sensation of accomplishment my character is enjoying in game. I didn't really slay a bunch of orcs in real life, but it was hard work to get through it on paper and my character didn't die in the end, so it feels like I'm being rewarded when I can again say and do anything without waiting for my turn.
On the other hand, the structure of wargaming is a great luxury. The scope of a wargame is clear and easy to anticipate. It's easy to wargame solo, and it can include roleplay elements as you build and level-up an army. It's easy to track progress and game state, and I love that I can have a wargame set up on a table and play a round or two each day for a three or four days, and then spend some time updating stats, and proceed to the next battle.
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